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class="vector-toc-numb">2</span> <span>General formulation</span> </div> </a> <ul id="toc-General_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Special cases</span> </div> </a> <button aria-controls="toc-Special_cases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Special cases subsection</span> </button> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> <li id="toc-Two-body_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-body_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Two-body problem</span> </div> </a> <ul id="toc-Two-body_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Three-body_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three-body_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Three-body problem</span> </div> </a> <ul id="toc-Three-body_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Four-body_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Four-body_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Four-body problem</span> </div> </a> <ul id="toc-Four-body_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planetary_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Planetary_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Planetary problem</span> </div> </a> <ul id="toc-Planetary_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Central_configurations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Central_configurations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Central configurations</span> </div> </a> <ul id="toc-Central_configurations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n-body_choreography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n-body_choreography"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span><span>n</span>-body choreography</span> </div> </a> <ul id="toc-n-body_choreography-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analytic_approaches" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analytic_approaches"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Analytic approaches</span> </div> </a> <button aria-controls="toc-Analytic_approaches-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Analytic approaches subsection</span> </button> <ul id="toc-Analytic_approaches-sublist" class="vector-toc-list"> <li id="toc-Power_series_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_series_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Power series solution</span> </div> </a> <ul id="toc-Power_series_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_generalized_Sundman_global_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_generalized_Sundman_global_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>A generalized Sundman global solution</span> </div> </a> <ul id="toc-A_generalized_Sundman_global_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singularities_of_the_n-body_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singularities_of_the_n-body_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Singularities of the <span>n</span>-body problem</span> </div> </a> <ul id="toc-Singularities_of_the_n-body_problem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Simulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Simulation</span> </div> </a> <button aria-controls="toc-Simulation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Simulation subsection</span> </button> <ul id="toc-Simulation-sublist" class="vector-toc-list"> <li id="toc-Few_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Few_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Few bodies</span> </div> </a> <ul id="toc-Few_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Many_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Many_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Many bodies</span> </div> </a> <ul id="toc-Many_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strong_gravitation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strong_gravitation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Strong gravitation</span> </div> </a> <ul id="toc-Strong_gravitation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_n-body_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_n-body_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other <span>n</span>-body problems</span> </div> </a> <ul id="toc-Other_n-body_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> 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Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A3%D9%84%D8%A9_%D9%86_%D9%85%D9%86_%D8%A7%D9%84%D8%A3%D8%AC%D8%B3%D8%A7%D9%85" title="مسألة ن من الأجسام – Arabic" lang="ar" hreflang="ar" data-title="مسألة ن من الأجسام" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D1%96%D1%82%D0%B0%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_N_%D1%86%D0%B5%D0%BB" title="Гравітацыйная задача N цел – Belarusian" lang="be" hreflang="be" data-title="Гравітацыйная задача N цел" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Problema_dels_n_cossos" title="Problema dels n cossos – Catalan" lang="ca" hreflang="ca" data-title="Problema dels n cossos" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/N-K%C3%B6rper-Problem" title="N-Körper-Problem – German" lang="de" hreflang="de" data-title="N-Körper-Problem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Problema_de_los_n_cuerpos" title="Problema de los n cuerpos – Spanish" lang="es" hreflang="es" data-title="Problema de los n cuerpos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B3%D8%A6%D9%84%D9%87_(n)_%D8%AC%D8%B3%D9%85" title="مسئله (n) جسم – Persian" lang="fa" hreflang="fa" data-title="مسئله (n) جسم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Probl%C3%A8me_%C3%A0_N_corps" title="Problème à N corps – French" lang="fr" hreflang="fr" data-title="Problème à N corps" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%B2%B4_%EB%AC%B8%EC%A0%9C" title="다체 문제 – Korean" lang="ko" hreflang="ko" data-title="다체 문제" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Problema_degli_n-corpi" title="Problema degli n-corpi – Italian" lang="it" hreflang="it" data-title="Problema degli n-corpi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%A2%D7%99%D7%94_n-%D7%92%D7%95%D7%A4%D7%99%D7%AA" title="בעיה n-גופית – Hebrew" lang="he" hreflang="he" data-title="בעיה n-גופית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N-test_probl%C3%A9ma" title="N-test probléma – Hungarian" lang="hu" hreflang="hu" data-title="N-test probléma" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E4%BD%93%E5%95%8F%E9%A1%8C" title="多体問題 – Japanese" lang="ja" hreflang="ja" data-title="多体問題" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Problem_n_cia%C5%82" title="Problem n ciał – Polish" lang="pl" hreflang="pl" data-title="Problem n ciał" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Problema_dos_n-corpos" title="Problema dos n-corpos – Portuguese" lang="pt" hreflang="pt" data-title="Problema dos n-corpos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_N_%D1%82%D0%B5%D0%BB" title="Гравитационная задача N тел – Russian" lang="ru" hreflang="ru" data-title="Гравитационная задача N тел" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Probl%C3%A9m_n_telies" title="Problém n telies – Slovak" lang="sk" hreflang="sk" data-title="Problém n telies" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9B%E0%B8%B1%E0%B8%8D%E0%B8%AB%E0%B8%B2%E0%B8%AB%E0%B8%A5%E0%B8%B2%E0%B8%A2%E0%B8%A7%E0%B8%B1%E0%B8%95%E0%B8%96%E0%B8%B8" title="ปัญหาหลายวัตถุ – Thai" lang="th" hreflang="th" data-title="ปัญหาหลายวัตถุ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D1%96%D1%82%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D0%B0_%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_N_%D1%82%D1%96%D0%BB" title="Гравітаційна задача N тіл – Ukrainian" lang="uk" hreflang="uk" data-title="Гравітаційна задача N тіл" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/N%E4%BD%93%E9%97%AE%E9%A2%98" title="N体问题 – Chinese" lang="zh" hreflang="zh" data-title="N体问题" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1199050#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> 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searchaux" style="display:none">Problem in physics and celestial mechanics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the problem in classical mechanics. For the problem in quantum mechanics, see <a href="/wiki/Many-body_problem" title="Many-body problem">Many-body problem</a>. For engineering problems and simulations involving many components, see <a href="/wiki/Multibody_system" title="Multibody system">Multibody system</a> and <a href="/wiki/Multibody_simulation" title="Multibody simulation">Multibody simulation</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle">Astrodynamics</th></tr><tr><td class="sidebar-image" style="padding-bottom:0.85em;"><span typeof="mw:File"><a href="/wiki/File:Orbit_mechanics_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/60px-Orbit_mechanics_icon.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/90px-Orbit_mechanics_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/120px-Orbit_mechanics_icon.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td></tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics"><span style="font-size:110%;">Orbital mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Apsis" title="Apsis">Apsis</a></li> <li><a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><a href="/wiki/Orbital_node" title="Orbital node">Orbital nodes</a></li> <li><a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Types of <a href="/wiki/Two-body_problem" title="Two-body problem">two-body orbits</a> by <br />eccentricity</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular orbit</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptic orbit</a></li></ul> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Transfer_orbit" title="Transfer orbit">Transfer orbit</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer orbit</a></li><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer orbit</a>)</li></ul></div></div> <ul><li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic orbit</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial orbit</a></li> <li><a href="/wiki/Orbital_decay" title="Orbital decay">Decaying orbit</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Dynamical_friction" title="Dynamical friction">Dynamical friction</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Kepler%27s_equation" title="Kepler's equation">Kepler's equation</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><a href="/wiki/Orbital_speed" title="Orbital speed">Orbital velocity</a></li> <li><a href="/wiki/Surface_gravity" title="Surface gravity">Surface gravity</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Vis-viva_equation" title="Vis-viva equation">Vis-viva equation</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Celestial_mechanics" title="Celestial mechanics"><span style="font-size:110%;">Celestial mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Gravitational influences</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbations</a></li> <li><a href="/wiki/Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">Sphere of influence</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a class="mw-selflink selflink">N-body orbits</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian points</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Halo_orbit" title="Halo orbit">Halo orbits</a>)</li></ul></div></div> <ul><li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous orbits</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov orbits</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Aerospace_engineering" title="Aerospace engineering"><span style="font-size:110%;">Engineering and efficiency</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Preflight engineering</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Mass_ratio" title="Mass ratio">Mass ratio</a></li> <li><a href="/wiki/Payload_fraction" title="Payload fraction">Payload fraction</a></li> <li><a href="/wiki/Propellant_mass_fraction" title="Propellant mass fraction">Propellant mass fraction</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Efficiency measures</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Propulsive maneuvers</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Orbital maneuver</a></li> <li><a href="/wiki/Orbit_insertion" title="Orbit insertion">Orbit insertion</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Astrodynamics" title="Template:Astrodynamics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Astrodynamics" title="Template talk:Astrodynamics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Astrodynamics" title="Special:EditPage/Template:Astrodynamics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b><span class="texhtml mvar" style="font-style:italic;">n</span>-body problem</b> is the problem of predicting the individual motions of a group of <a href="/wiki/Astronomical_object" title="Astronomical object">celestial objects</a> interacting with each other <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitationally</a>.<sup id="cite_ref-Leimanis_and_Minorsky_1-0" class="reference"><a href="#cite_note-Leimanis_and_Minorsky-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Solving this problem has been motivated by the desire to understand the motions of the <a href="/wiki/Sun" title="Sun">Sun</a>, <a href="/wiki/Moon" title="Moon">Moon</a>, <a href="/wiki/Planet" title="Planet">planets</a>, and visible <a href="/wiki/Star" title="Star">stars</a>. In the 20th century, understanding the dynamics of <a href="/wiki/Globular_cluster" title="Globular cluster">globular cluster</a> star systems became an important <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem.<sup id="cite_ref-Heggie_and_Hut_2-0" class="reference"><a href="#cite_note-Heggie_and_Hut-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem in <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is considerably more difficult to solve due to additional factors like time and space distortions. </p><p>The classical physical problem can be informally stated as the following: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Given the quasi-steady orbital properties (instantaneous position, velocity and time)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>The <a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a> has been completely solved and is discussed below, as well as the famous <i>restricted</i> <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Knowing three orbital positions of a planet's orbit – positions obtained by Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> from astronomer <a href="/wiki/John_Flamsteed" title="John Flamsteed">John Flamsteed</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits correctly or even very well.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Newton realized that this was because gravitational interactive forces amongst all the planets were affecting all their orbits. </p><p>The aforementioned revelation strikes directly at the core of what the n-body issue physically is: as Newton understood, it is not enough to just provide the beginning location and velocity, or even three orbital positions, in order to establish a planet's actual orbit; one must also be aware of the gravitational interaction forces. Thus came the awareness and rise of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body "problem" in the early 17th century. These gravitational attractive forces do conform to Newton's laws of motion and to his law of universal gravitation, but the many multiple (<span class="texhtml mvar" style="font-style:italic;">n</span>-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach. </p><p>After Newton's time the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem is unsolvable because of those gravitational interactive forces.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Newton said<sup id="cite_ref-Cohen_10-0" class="reference"><a href="#cite_note-Cohen-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in his <i>Principia</i>, paragraph 21: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>And hence it is that the attractive force is found in both bodies. The Sun attracts Jupiter and the other planets, Jupiter attracts its satellites and similarly the satellites act on one another. And although the actions of each of a pair of planets on the other can be distinguished from each other and can be considered as two actions by which each attracts the other, yet inasmuch as they are between the same, two bodies they are not two but a simple operation between two termini. Two bodies can be drawn to each other by the contraction of rope between them. The cause of the action is twofold, namely the disposition of each of the two bodies; the action is likewise twofold, insofar as it is upon two bodies; but insofar as it is between two bodies it is single and one ...</p></blockquote> <p>Newton concluded via his <a href="/wiki/Newton%27s_third_law" class="mw-redirect" title="Newton's third law">third law of motion</a> that "according to this Law all bodies must attract each other." This last statement, which implies the existence of gravitational interactive forces, is key. </p><p>As shown below, the problem also conforms to <a href="/wiki/Jean_Le_Rond_D%27Alembert" class="mw-redirect" title="Jean Le Rond D'Alembert">Jean Le Rond D'Alembert</a>'s non-Newtonian first and second Principles and to the nonlinear <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces. </p><p>The problem of finding the general solution of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem was considered very important and challenging. Indeed, in the late 19th century King <a href="/wiki/Oscar_II_of_Sweden" class="mw-redirect" title="Oscar II of Sweden">Oscar II of Sweden</a>, advised by <a href="/wiki/G%C3%B6sta_Mittag-Leffler" title="Gösta Mittag-Leffler">Gösta Mittag-Leffler</a>, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series <i>converges uniformly</i>.</p></blockquote> <p>In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a>, even though he did not solve the original problem. (The first version of his contribution even contained a serious error.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>) The version finally printed contained many important ideas which led to the development of <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>. The problem as stated originally was finally solved by <a href="/wiki/Karl_F._Sundman" title="Karl F. Sundman">Karl Fritiof Sundman</a> for <span class="texhtml"><i>n</i> = 3</span> and generalized to <span class="texhtml"><i>n</i> > 3</span> by L. K. Babadzanjanz<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Qiudong_Wang" title="Qiudong Wang">Qiudong Wang</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="General_formulation">General formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=2" title="Edit section: General formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem considers <span class="texhtml mvar" style="font-style:italic;">n</span> point masses <span class="texhtml"><i>m<sub>i</sub></i>, <i>i</i> = 1, 2, …, <i>n</i></span> in an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial reference frame</a> in three dimensional space <span class="texhtml">ℝ<sup>3</sup></span> moving under the influence of mutual gravitational attraction. Each mass <span class="texhtml"><i>m<sub>i</sub></i></span> has a position vector <span class="texhtml"><b>q</b><sub><i>i</i></sub></span>. <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> says that mass times acceleration <span class="texhtml"><i>m<sub>i</sub></i> <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>2</sup><b>q</b><sub><i>i</i></sub></span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>2</sup></span></span>⁠</span></span> is equal to the sum of the forces on the mass. <a href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</a> says that the gravitational force felt on mass <span class="texhtml"><i>m<sub>i</sub></i></span> by a single mass <span class="texhtml"><i>m<sub>j</sub></i></span> is given by<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{ij}={\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}}}\cdot {\frac {\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}={\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{ij}={\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}}}\cdot {\frac {\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}={\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c5aab28749b00eb610136b76689a0f6cf57976" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:52.144ex; height:8.009ex;" alt="{\displaystyle \mathbf {F} _{ij}={\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}}}\cdot {\frac {\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}={\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}},}"></span> where <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a> and <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><b>q</b><sub><i>j</i></sub> − <b>q</b><sub><i>i</i></sub></span>‖</span> is the magnitude of the distance between <span class="texhtml"><b>q</b><sub><i>i</i></sub></span> and <span class="texhtml"><b>q</b><sub><i>j</i></sub></span> (<a href="/wiki/Metric_(mathematics)#Metrics_on_vector_spaces" class="mw-redirect" title="Metric (mathematics)">metric induced by</a> the <a href="/wiki/Norm_(mathematics)#Taxicab_norm_or_Manhattan_norm" title="Norm (mathematics)"><span class="texhtml"><i>l</i><sub>2</sub></span> norm</a>). </p><p> Summing over all masses yields the <span class="texhtml mvar" style="font-style:italic;">n</span>-body <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a>:</p><div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}{\frac {d^{2}\mathbf {q} _{i}}{dt^{2}}}=\sum _{j=1 \atop j\neq i}^{n}{\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}}=-{\frac {\partial U}{\partial \mathbf {q} _{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}{\frac {d^{2}\mathbf {q} _{i}}{dt^{2}}}=\sum _{j=1 \atop j\neq i}^{n}{\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}}=-{\frac {\partial U}{\partial \mathbf {q} _{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95000ab60f0bec4d3d3f5e2785e59ee32e826a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:42.613ex; height:9.009ex;" alt="{\displaystyle m_{i}{\frac {d^{2}\mathbf {q} _{i}}{dt^{2}}}=\sum _{j=1 \atop j\neq i}^{n}{\frac {Gm_{i}m_{j}\left(\mathbf {q} _{j}-\mathbf {q} _{i}\right)}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{3}}}=-{\frac {\partial U}{\partial \mathbf {q} _{i}}}}"></span> </p> </div><p>where <span class="texhtml mvar" style="font-style:italic;">U</span> is the <i>self-potential</i> energy </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=-\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98af3a8bb6f5ee271b5b3e486e94fcc638201b3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.532ex; height:7.176ex;" alt="{\displaystyle U=-\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|}}.}"></span> </p><p>Defining the momentum to be <span class="texhtml"><b>p</b><sub><i>i</i></sub> = <i>m<sub>i</sub></i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><b>q</b><sub><i>i</i></sub></span><span class="sr-only">/</span><span class="den"><i>dt</i></span></span>⁠</span></span>, <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamilton's equations of motion</a> for the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem become<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {q} _{i}}{dt}}={\frac {\partial H}{\partial \mathbf {p} _{i}}}\qquad {\frac {d\mathbf {p} _{i}}{dt}}=-{\frac {\partial H}{\partial \mathbf {q} _{i}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {q} _{i}}{dt}}={\frac {\partial H}{\partial \mathbf {p} _{i}}}\qquad {\frac {d\mathbf {p} _{i}}{dt}}=-{\frac {\partial H}{\partial \mathbf {q} _{i}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1566d6a27f227a87127661a8eca8ce497951be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.701ex; height:6.176ex;" alt="{\displaystyle {\frac {d\mathbf {q} _{i}}{dt}}={\frac {\partial H}{\partial \mathbf {p} _{i}}}\qquad {\frac {d\mathbf {p} _{i}}{dt}}=-{\frac {\partial H}{\partial \mathbf {q} _{i}}},}"></span> where the <a href="/wiki/Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian function</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=T+U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=T+U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a84dbc508706ece6d41d942fa0e9751e016f3fa0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.421ex; height:2.343ex;" alt="{\displaystyle H=T+U}"></span> and <span class="texhtml mvar" style="font-style:italic;">T</span> is the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\sum _{i=1}^{n}{\frac {\left\|\mathbf {p} _{i}\right\|^{2}}{2m_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\sum _{i=1}^{n}{\frac {\left\|\mathbf {p} _{i}\right\|^{2}}{2m_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2add519be3f227f33aea508d4e90261a5b18414" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.624ex; height:7.176ex;" alt="{\displaystyle T=\sum _{i=1}^{n}{\frac {\left\|\mathbf {p} _{i}\right\|^{2}}{2m_{i}}}.}"></span> </p><p>Hamilton's equations show that the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem is a system of <span class="texhtml">6<i>n</i></span> first-order <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, with <span class="texhtml">6<i>n</i></span> <a href="/wiki/Initial_condition" title="Initial condition">initial conditions</a> as <span class="texhtml">3<i>n</i></span> initial position coordinates and <span class="texhtml">3<i>n</i></span> initial momentum values. </p><p>Symmetries in the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem yield global <a href="/wiki/Integral_of_motion" class="mw-redirect" title="Integral of motion">integrals of motion</a> that simplify the problem.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Translational_symmetry" title="Translational symmetry">Translational symmetry</a> of the problem results in the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ={\frac {\displaystyle \sum _{i=1}^{n}m_{i}\mathbf {q} _{i}}{\displaystyle \sum _{i=1}^{n}m_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ={\frac {\displaystyle \sum _{i=1}^{n}m_{i}\mathbf {q} _{i}}{\displaystyle \sum _{i=1}^{n}m_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ffab53dbbe80ea26e784fb55f6a1c13046914a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:14.659ex; height:14.509ex;" alt="{\displaystyle \mathbf {C} ={\frac {\displaystyle \sum _{i=1}^{n}m_{i}\mathbf {q} _{i}}{\displaystyle \sum _{i=1}^{n}m_{i}}}}"></span> moving with constant velocity, so that <span class="texhtml"><b>C</b> = <b>L</b><sub>0</sub><i>t</i> + <b>C</b><sub>0</sub></span>, where <span class="texhtml"><b>L</b><sub>0</sub></span> is the linear velocity and <span class="texhtml"><b>C</b><sub>0</sub></span> is the initial position. The constants of motion <span class="texhtml"><b>L</b><sub>0</sub></span> and <span class="texhtml"><b>C</b><sub>0</sub></span> represent six integrals of the motion. <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">Rotational symmetry</a> results in the total <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> being constant <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =\sum _{i=1}^{n}\mathbf {q} _{i}\times \mathbf {p} _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =\sum _{i=1}^{n}\mathbf {q} _{i}\times \mathbf {p} _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516d2d35c82f4373b92a0cb35777555a94de0168" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.843ex; height:6.843ex;" alt="{\displaystyle \mathbf {A} =\sum _{i=1}^{n}\mathbf {q} _{i}\times \mathbf {p} _{i},}"></span> where × is the <a href="/wiki/Cross_product" title="Cross product">cross product</a>. The three components of the total angular momentum <span class="texhtml"><b>A</b></span> yield three more constants of the motion. The last general constant of the motion is given by the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> <span class="texhtml mvar" style="font-style:italic;">H</span>. Hence, every <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem has ten integrals of motion. </p><p>Because <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">U</span> are <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous functions</a> of degree 2 and −1, respectively, the equations of motion have a <a href="/wiki/Scaling_invariance" class="mw-redirect" title="Scaling invariance">scaling invariance</a>: if <span class="texhtml"><b>q</b><sub><i>i</i></sub>(<i>t</i>)</span> is a solution, then so is <span class="texhtml"><i>λ</i><sup>−2/3</sup><b>q</b><sub><i>i</i></sub>(<i>λt</i>)</span> for any <span class="texhtml"><i>λ</i> > 0</span>.<sup id="cite_ref-Chenciner_2007_18-0" class="reference"><a href="#cite_note-Chenciner_2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> of an <span class="texhtml mvar" style="font-style:italic;">n</span>-body system is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\sum _{i=1}^{n}m_{i}\mathbf {q} _{i}\cdot \mathbf {q} _{i}=\sum _{i=1}^{n}m_{i}\left\|\mathbf {q} _{i}\right\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\sum _{i=1}^{n}m_{i}\mathbf {q} _{i}\cdot \mathbf {q} _{i}=\sum _{i=1}^{n}m_{i}\left\|\mathbf {q} _{i}\right\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac773f4de75cb30eb8a7cc4c3822b66538f8045" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.223ex; height:6.843ex;" alt="{\displaystyle I=\sum _{i=1}^{n}m_{i}\mathbf {q} _{i}\cdot \mathbf {q} _{i}=\sum _{i=1}^{n}m_{i}\left\|\mathbf {q} _{i}\right\|^{2}}"></span> and the <i>virial</i> is given by <span class="texhtml"><i>Q</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dI</i></span><span class="sr-only">/</span><span class="den"><i>dt</i></span></span>⁠</span></span>. Then the <i>Lagrange–Jacobi formula</i> states that<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}I}{dt^{2}}}=2T-U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>T</mi> <mo>−</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}I}{dt^{2}}}=2T-U.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257156ba5b19a58790e1e7078cef2b82ed516e1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.447ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}I}{dt^{2}}}=2T-U.}"></span> </p><p>For systems in <i>dynamic equilibrium</i>, the longterm time average of <span class="texhtml">⟨<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>2</sup><i>I</i></span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>2</sup></span></span>⁠</span>⟩</span> is zero. Then on average the total kinetic energy is half the total potential energy, <span class="texhtml">⟨<i>T</i>⟩ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>⟨<i>U</i>⟩</span>, which is an example of the <a href="/wiki/Virial_theorem" title="Virial theorem">virial theorem</a> for gravitational systems.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> If <span class="texhtml mvar" style="font-style:italic;">M</span> is the total mass and <span class="texhtml mvar" style="font-style:italic;">R</span> a characteristic size of the system (for example, the radius containing half the mass of the system), then the critical time for a system to settle down to a dynamic equilibrium is<sup id="cite_ref-Trenti_2008_21-0" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\mathrm {cr} }={\sqrt {\frac {GM}{R^{3}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\mathrm {cr} }={\sqrt {\frac {GM}{R^{3}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3505fdb2f6e57a7419f61c059b5b54a7fdf5740f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.621ex; height:7.676ex;" alt="{\displaystyle t_{\mathrm {cr} }={\sqrt {\frac {GM}{R^{3}}}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=3" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Two-body_problem">Two-body problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=4" title="Edit section: Two-body problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Two-body_problem" title="Two-body problem">Two-body problem</a></div> <p>Any discussion of planetary interactive forces has always started historically with the two-body problem. The purpose of this section is to relate the real complexity in calculating any planetary forces. Note in this Section also, several subjects, such as <a href="/wiki/Gravity" title="Gravity">gravity</a>, <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a>, <a href="/wiki/Kepler%27s_Laws" class="mw-redirect" title="Kepler's Laws">Kepler's Laws</a>, etc.; and in the following Section too (<a href="/wiki/Three-body_problem" title="Three-body problem">Three-body problem</a>) are discussed on other Wikipedia pages. Here though, these subjects are discussed from the perspective of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem. </p><p>The two-body problem (<span class="texhtml"><i>n</i> = 2</span>) was completely solved by <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a> (1667–1748) by classical theory (and not by Newton) by assuming the main point-mass was fixed; this is outlined here.<sup id="cite_ref-Bate,_Mueller,_and_White_22-0" class="reference"><a href="#cite_note-Bate,_Mueller,_and_White-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Consider then the motion of two bodies, say the Sun and the Earth, with the Sun fixed, then: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}m_{1}\mathbf {a} _{1}&={\frac {Gm_{1}m_{2}}{r_{12}^{3}}}(\mathbf {r} _{2}-\mathbf {r} _{1})&&\quad {\text{Sun–Earth}}\\m_{2}\mathbf {a} _{2}&={\frac {Gm_{1}m_{2}}{r_{21}^{3}}}(\mathbf {r} _{1}-\mathbf {r} _{2})&&\quad {\text{Earth–Sun}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sun–Earth</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>Earth–Sun</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m_{1}\mathbf {a} _{1}&={\frac {Gm_{1}m_{2}}{r_{12}^{3}}}(\mathbf {r} _{2}-\mathbf {r} _{1})&&\quad {\text{Sun–Earth}}\\m_{2}\mathbf {a} _{2}&={\frac {Gm_{1}m_{2}}{r_{21}^{3}}}(\mathbf {r} _{1}-\mathbf {r} _{2})&&\quad {\text{Earth–Sun}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7d77a34897dcdf0ccfa89e6023af6c4f5dbf13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:44.975ex; height:13.009ex;" alt="{\displaystyle {\begin{aligned}m_{1}\mathbf {a} _{1}&={\frac {Gm_{1}m_{2}}{r_{12}^{3}}}(\mathbf {r} _{2}-\mathbf {r} _{1})&&\quad {\text{Sun–Earth}}\\m_{2}\mathbf {a} _{2}&={\frac {Gm_{1}m_{2}}{r_{21}^{3}}}(\mathbf {r} _{1}-\mathbf {r} _{2})&&\quad {\text{Earth–Sun}}\end{aligned}}}"></span> </p><p>The equation describing the motion of mass <span class="texhtml"><i>m</i><sub>2</sub></span> relative to mass <span class="texhtml"><i>m</i><sub>1</sub></span> is readily obtained from the differences between these two equations and after canceling common terms gives: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\alpha } +{\frac {\eta }{r^{3}}}\mathbf {r} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\alpha } +{\frac {\eta }{r^{3}}}\mathbf {r} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3946d49f753720a5c6b6fb23c21ae02a120035" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.804ex; height:5.176ex;" alt="{\displaystyle \mathbf {\alpha } +{\frac {\eta }{r^{3}}}\mathbf {r} =\mathbf {0} }"></span> Where </p> <ul><li><span class="texhtml"><b>r</b> = <b>r</b><sub>2</sub> − <b>r</b><sub>1</sub></span> is the vector position of <span class="texhtml"><i>m</i><sub>2</sub></span> relative to <span class="texhtml"><i>m</i><sub>1</sub></span>;</li> <li><span class="texhtml mvar" style="font-style:italic;">α</span> is the <i>Eulerian</i> acceleration <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>2</sup><b>r</b></span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>2</sup></span></span>⁠</span></span>;</li> <li><span class="texhtml"><i>η</i> = <i>G</i>(<i>m</i><sub>1</sub> + <i>m</i><sub>2</sub>)</span>.</li></ul> <p>The equation <span class="texhtml"><i>α</i> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>η</i></span><span class="sr-only">/</span><span class="den"><i>r</i><sup>3</sup></span></span>⁠</span><b>r</b> = 0</span> is the fundamental differential equation for the two-body problem Bernoulli solved in 1734. Notice for this approach forces have to be determined first, then the equation of motion resolved. This differential equation has elliptic, or parabolic or hyperbolic solutions.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>It is incorrect to think of <span class="texhtml"><i>m</i><sub>1</sub></span> (the Sun) as fixed in space when applying Newton's law of universal gravitation, and to do so leads to erroneous results. The fixed point for two isolated gravitationally interacting bodies is their mutual <a href="/wiki/Barycentric_coordinates_(astronomy)" class="mw-redirect" title="Barycentric coordinates (astronomy)">barycenter</a>, and this <a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a> can be solved exactly, such as using <a href="/wiki/Jacobi_coordinates" title="Jacobi coordinates">Jacobi coordinates</a> relative to the barycenter. </p><p>Dr. Clarence Cleminshaw calculated the approximate position of the Solar System's barycenter, a result achieved mainly by combining only the masses of Jupiter and the Sun. <i>Science Program</i> stated in reference to his work: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The Sun contains 98 per cent of the mass in the solar system, with the superior planets beyond Mars accounting for most of the rest. On the average, the center of the mass of the Sun–Jupiter system, when the two most massive objects are considered alone, lies 462,000 miles from the Sun's center, or some 30,000 miles above the solar surface! Other large planets also influence the center of mass of the solar system, however. In 1951, for example, the systems' center of mass was not far from the Sun's center because Jupiter was on the opposite side from Saturn, Uranus and Neptune. In the late 1950s, when all four of these planets were on the same side of the Sun, the system's center of mass was more than 330,000 miles from the solar surface, Dr. C. H. Cleminshaw of Griffith Observatory in Los Angeles has calculated.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></p></blockquote> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kipler%27s_Error.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Kipler%27s_Error.jpg/220px-Kipler%27s_Error.jpg" decoding="async" width="220" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Kipler%27s_Error.jpg/330px-Kipler%27s_Error.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/13/Kipler%27s_Error.jpg 2x" data-file-width="375" data-file-height="455" /></a><figcaption>Real motion versus Kepler's apparent motion</figcaption></figure> <p>The Sun wobbles as it rotates around the <a href="/wiki/Galactic_Center" title="Galactic Center">Galactic Center</a>, dragging the Solar System and Earth along with it. What mathematician <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a> did in arriving at his three famous equations was curve-fit the apparent motions of the planets using <a href="/wiki/Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a>'s data, and not curve-fitting their true circular motions about the Sun (see Figure). Both <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> and Newton were well aware that Newton's <i>Law of Universal Gravitation</i> did not hold for the forces associated with elliptical orbits.<sup id="cite_ref-Cohen_10-1" class="reference"><a href="#cite_note-Cohen-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In fact, Newton's Universal Law does not account for the orbit of Mercury, the asteroid belt's gravitational behavior, or <a href="/wiki/Rings_of_Saturn" title="Rings of Saturn">Saturn's rings</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Newton stated (in section 11 of the <i>Principia</i>) that the main reason, however, for failing to predict the forces for elliptical orbits was that his math model was for a body confined to a situation that hardly existed in the real world, namely, the motions of bodies attracted toward an unmoving center. Some present physics and astronomy textbooks do not emphasize the negative significance of Newton's assumption and end up teaching that his mathematical model is in effect reality. It is to be understood that the classical two-body problem solution above is a mathematical idealization. See also <a href="/wiki/Kepler%27s_laws_of_planetary_motion#Kepler's_first_law" title="Kepler's laws of planetary motion">Kepler's first law of planetary motion</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Three-body_problem">Three-body problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=5" title="Edit section: Three-body problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Three-body_problem" title="Three-body problem">Three-body problem</a></div> <p>This section relates a historically important <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem solution after simplifying assumptions were made. </p><p>In the past not much was known about the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem for <span class="texhtml"><i>n</i> ≥ 3</span>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The case <span class="texhtml"><i>n</i> = 3</span> has been the most studied. Many earlier attempts to understand the three-body problem were quantitative, aiming at finding explicit solutions for special situations. </p> <ul><li>In 1687, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> published in the <i>Principia</i> the first steps in the study of the problem of the movements of three bodies subject to their mutual gravitational attractions, but his efforts resulted in verbal descriptions and geometrical sketches; see especially Book 1, Proposition 66 and its corollaries (Newton, 1687 and 1999 (transl.), see also Tisserand, 1894).</li> <li>In 1767, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> found <a href="/wiki/Collinear" class="mw-redirect" title="Collinear">collinear</a> motions, in which three bodies of any masses move proportionately along a fixed straight line. The <a href="/wiki/Euler%27s_three-body_problem" title="Euler's three-body problem">Euler's three-body problem</a> is the special case in which two of the bodies are fixed in space (this should not be confused with the <a href="/wiki/Circular_restricted_three-body_problem" class="mw-redirect" title="Circular restricted three-body problem">circular restricted three-body problem</a>, in which the two massive bodies describe a circular orbit and are only fixed in a synodic reference frame).</li> <li>In 1772, <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a> discovered two classes of periodic solution, each for three bodies of any masses. In one class, the bodies lie on a rotating straight line. In the other class, the bodies lie at the vertices of a rotating equilateral triangle. In either case, the paths of the bodies will be conic sections. Those solutions led to the study of <i>central configurations</i>, for which <span class="texhtml"><i>q̈</i> = <i>kq</i></span> for some constant <span class="texhtml"><i>k</i> > 0</span>.</li> <li>A major study of the Earth–Moon–Sun system was undertaken by <a href="/wiki/Charles-Eug%C3%A8ne_Delaunay" title="Charles-Eugène Delaunay">Charles-Eugène Delaunay</a>, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>.</li> <li>In 1917, <a href="/wiki/Forest_Ray_Moulton" title="Forest Ray Moulton">Forest Ray Moulton</a> published his now classic, <i>An Introduction to Celestial Mechanics</i> (see references) with its plot of the <i>restricted three-body problem</i> solution (see figure below).<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> An aside, see Meirovitch's book, pages 413–414 for his restricted three-body problem solution.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:N-body_problem_(3).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f9/N-body_problem_%283%29.gif" decoding="async" width="182" height="174" class="mw-file-element" data-file-width="182" data-file-height="174" /></a><figcaption>Motion of three particles under gravity, demonstrating chaotic behaviour</figcaption></figure> <p>Moulton's solution may be easier to visualize (and definitely easier to solve) if one considers the more massive body (such as the <a href="/wiki/Sun" title="Sun">Sun</a>) to be stationary in space, and the less massive body (such as <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>) to orbit around it, with the equilibrium points (<a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian points</a>) maintaining the 60° spacing ahead of, and behind, the less massive body almost in its orbit (although in reality neither of the bodies are truly stationary, as they both orbit the center of mass of the whole system—about the barycenter). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrangian points. See figure below: </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Restricted_3-Body_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Restricted_3-Body_1.jpg/220px-Restricted_3-Body_1.jpg" decoding="async" width="220" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Restricted_3-Body_1.jpg/330px-Restricted_3-Body_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Restricted_3-Body_1.jpg/440px-Restricted_3-Body_1.jpg 2x" data-file-width="525" data-file-height="495" /></a><figcaption>Restricted three-body problem</figcaption></figure> <p>In the <i>restricted three-body problem</i> math model figure above (after Moulton), the Lagrangian points L<sub>4</sub> and L<sub>5</sub> are where the <a href="/wiki/Trojan_(astronomy)" class="mw-redirect" title="Trojan (astronomy)">Trojan</a> planetoids resided (see <a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian point</a>); <span class="texhtml"><i>m</i><sub>1</sub></span> is the Sun and <span class="texhtml"><i>m</i><sub>2</sub></span> is Jupiter. L<sub>2</sub> is a point within the asteroid belt. It has to be realized for this model, this whole Sun-Jupiter diagram is rotating about its barycenter. The restricted three-body problem solution predicted the Trojan planetoids before they were first seen. The <span class="texhtml mvar" style="font-style:italic;">h</span>-circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter. It is conjectured, contrary to Richard H. Batin's conjecture (see References), the two <span class="texhtml"><i>h</i><sub>1</sub></span> are gravity sinks, in and where gravitational forces are zero, and the reason the Trojan planetoids are trapped there. The total amount of mass of the planetoids is unknown. </p><p>The restricted three-body problem assumes the <a href="/wiki/Mass" title="Mass">mass</a> of one of the bodies is negligible.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see <a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a>; for binary systems, see <a href="/wiki/Roche_lobe" title="Roche lobe">Roche lobe</a>. Specific solutions to the three-body problem result in chaotic motion with no obvious sign of a repetitious path.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> </p><p>The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, most notably by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of <a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> chaos theory.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> In the restricted problem, there exist five <a href="/wiki/Equilibrium_points" class="mw-redirect" title="Equilibrium points">equilibrium points</a>. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. </p> <div class="mw-heading mw-heading3"><h3 id="Four-body_problem">Four-body problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=6" title="Edit section: Four-body problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Inspired by the circular restricted three-body problem, the four-body problem can be greatly simplified by considering a smaller body to have a small mass compared to the other three massive bodies, which in turn are approximated to describe circular orbits. This is known as the bicircular restricted four-body problem (also known as bicircular model) and it can be traced back to 1960 in a NASA report written by Su-Shu Huang.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> This formulation has been highly relevant in the <a href="/wiki/Astrodynamics" class="mw-redirect" title="Astrodynamics">astrodynamics</a>, mainly to model spacecraft trajectories in the Earth-Moon system with the addition of the gravitational attraction of the Sun. The former formulation of the bicircular restricted four-body problem can be problematic when modelling other systems than the Earth-Moon-Sun, so the formulation was generalized by Negri and Prado<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> to expand the application range and improve the accuracy without loss of simplicity. </p> <div class="mw-heading mw-heading3"><h3 id="Planetary_problem">Planetary problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=7" title="Edit section: Planetary problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>planetary problem</i> is the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem in the case that one of the masses is much larger than all the others. A prototypical example of a planetary problem is the Sun–<a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>–<a href="/wiki/Saturn" title="Saturn">Saturn</a> system, where the mass of the Sun is about 1000 times larger than the masses of Jupiter or Saturn.<sup id="cite_ref-Chenciner_2007_18-1" class="reference"><a href="#cite_note-Chenciner_2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> An approximate solution to the problem is to decompose it into <span class="texhtml"><i>n</i> − 1</span> pairs of star–planet <a href="/wiki/Kepler_problem" title="Kepler problem">Kepler problems</a>, treating interactions among the planets as perturbations. <a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbative approximation</a> works well as long as there are no <a href="/wiki/Orbital_resonance" title="Orbital resonance">orbital resonances</a> in the system, that is none of the ratios of unperturbed Kepler frequencies is a rational number. Resonances appear as small denominators in the expansion. </p><p>The existence of resonances and small denominators led to the important question of stability in the planetary problem: do planets, in nearly circular orbits around a star, remain in stable or bounded orbits over time?<sup id="cite_ref-Chenciner_2007_18-2" class="reference"><a href="#cite_note-Chenciner_2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chierchia_2010_33-0" class="reference"><a href="#cite_note-Chierchia_2010-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In 1963, <a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Vladimir Arnold</a> proved using <a href="/wiki/KAM_theory" class="mw-redirect" title="KAM theory">KAM theory</a> a kind of stability of the planetary problem: there exists a set of positive measure of <a href="/wiki/Quasiperiodic_motion" title="Quasiperiodic motion">quasiperiodic</a> orbits in the case of the planetary problem restricted to the plane.<sup id="cite_ref-Chierchia_2010_33-1" class="reference"><a href="#cite_note-Chierchia_2010-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In the KAM theory, chaotic planetary orbits would be bounded by quasiperiodic KAM tori. Arnold's result was extended to a more general theorem by Féjoz and Herman in 2004.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Central_configurations">Central configurations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=8" title="Edit section: Central configurations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Central_configuration" title="Central configuration">Central configuration</a></div> <p>A <a href="/wiki/Central_configuration" title="Central configuration">central configuration</a> <span class="texhtml"><b>q</b><sub>1</sub>(0), …, <b>q</b><sub><i>N</i></sub>(0)</span> is an initial configuration such that if the particles were all released with zero velocity, they would all collapse toward the center of mass <span class="texhtml"><b>C</b></span>.<sup id="cite_ref-Chierchia_2010_33-2" class="reference"><a href="#cite_note-Chierchia_2010-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Such a motion is called <i>homothetic</i>. Central configurations may also give rise to <i>homographic motions</i> in which all masses moves along Keplerian trajectories (elliptical, circular, parabolic, or hyperbolic), with all trajectories having the same eccentricity <span class="texhtml mvar" style="font-style:italic;">e</span>. For elliptical trajectories, <span class="texhtml"><i>e</i> = 1</span> corresponds to homothetic motion and <span class="texhtml"><i>e</i> = 0</span> gives a <i>relative equilibrium motion</i> in which the configuration remains an isometry of the initial configuration, as if the configuration was a rigid body.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Central configurations have played an important role in understanding the <a href="/wiki/Topology" title="Topology">topology</a> of <a href="/wiki/Invariant_manifold" title="Invariant manifold">invariant manifolds</a> created by fixing the first integrals of a system. </p> <div class="mw-heading mw-heading3"><h3 id="n-body_choreography"><span class="texhtml mvar" style="font-style:italic;">n</span>-body choreography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=9" title="Edit section: n-body choreography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/N-body_choreography" title="N-body choreography">n-body choreography</a></div> <p>Solutions in which all masses move on the <i>same</i> curve without collisions are called choreographies.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> A choreography for <span class="texhtml"><i>n</i> = 3</span> was discovered by Lagrange in 1772 in which three bodies are situated at the vertices of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> in the rotating frame. A <a href="/wiki/Lemniscate" title="Lemniscate">figure eight</a> choreography for <span class="texhtml"><i>n</i> = 3</span> was found numerically by C. Moore in 1993<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> and generalized and proven by A. Chenciner and R. Montgomery in 2000.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Since then, many other choreographies have been found for <span class="texhtml"><i>n</i> ≥ 3</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Analytic_approaches">Analytic approaches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=10" title="Edit section: Analytic approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For every solution of the problem, not only applying an <a href="/wiki/Isometry" title="Isometry">isometry</a> or a time shift but also a <a href="/wiki/T-symmetry" title="T-symmetry">reversal of time</a> (unlike in the case of friction) gives a solution as well.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> </p><p>In the physical literature about the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem (<span class="texhtml"><i>n</i> ≥ 3</span>), sometimes reference is made to "the impossibility of solving the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem" (via employing the above approach).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> However, care must be taken when discussing the 'impossibility' of a solution, as this refers only to the method of first integrals (compare the theorems by <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Abel</a> and <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois</a> about the impossibility of solving <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">algebraic equations of degree five</a> or higher by means of formulas only involving roots). </p> <div class="mw-heading mw-heading3"><h3 id="Power_series_solution">Power series solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=11" title="Edit section: Power series solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One way of solving the classical <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem is "the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem by <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>". </p><p>We start by defining the system of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2013)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\mathbf {x} _{i}(t)}{dt^{2}}}=G\sum _{k=1 \atop k\neq i}^{n}{\frac {m_{k}\left(\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right)}{\left|\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right|^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>G</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\mathbf {x} _{i}(t)}{dt^{2}}}=G\sum _{k=1 \atop k\neq i}^{n}{\frac {m_{k}\left(\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right)}{\left|\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right|^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65672aca11cd5a790f820a9427644526b2d23121" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.679ex; height:8.509ex;" alt="{\displaystyle {\frac {d^{2}\mathbf {x} _{i}(t)}{dt^{2}}}=G\sum _{k=1 \atop k\neq i}^{n}{\frac {m_{k}\left(\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right)}{\left|\mathbf {x} _{k}(t)-\mathbf {x} _{i}(t)\right|^{3}}},}"></span> </p><p>As <span class="texhtml"><b>x</b><sub><i>i</i></sub>(<i>t</i><sub>0</sub>)</span> and <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><b>x</b><sub><i>i</i></sub>(<i>t</i><sub>0</sub>)</span><span class="sr-only">/</span><span class="den"><i>dt</i></span></span>⁠</span></span> are given as initial conditions, every <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>2</sup><b>x</b><sub><i>i</i></sub>(<i>t</i>)</span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>2</sup></span></span>⁠</span></span> is known. Differentiating <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>2</sup><b>x</b><sub><i>i</i></sub>(<i>t</i>)</span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>2</sup></span></span>⁠</span></span> results in <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>d</i><sup>3</sup><b>x</b><sub><i>i</i></sub>(<i>t</i>)</span><span class="sr-only">/</span><span class="den"><i>dt</i><sup>3</sup></span></span>⁠</span></span> which at <span class="texhtml"><i>t</i><sub>0</sub></span> which is also known, and the Taylor series is constructed iteratively.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (March 2014)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="A_generalized_Sundman_global_solution">A generalized Sundman global solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=12" title="Edit section: A generalized Sundman global solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to generalize Sundman's result for the case <span class="texhtml"><i>n</i> > 3</span> (or <span class="texhtml"><i>n</i> = 3</span> and <span class="texhtml"><i>c</i> = 0</span><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="what is ''c''? (July 2012)">clarification needed</span></a></i>]</sup>) one has to face two obstacles: </p> <ol><li>As has been shown by Siegel, collisions which involve more than two bodies cannot be regularized analytically, hence Sundman's regularization cannot be generalized.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2017)">citation needed</span></a></i>]</sup></li> <li>The structure of singularities is more complicated in this case: other types of singularities may occur (see <a href="#Singularities_of_the_n-body_problem">below</a>).</li></ol> <p>Lastly, Sundman's result was generalized to the case of <span class="texhtml"><i>n</i> > 3</span> bodies by <a href="/wiki/Qiudong_Wang" title="Qiudong Wang">Qiudong Wang</a> in the 1990s.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities. The central point of his approach is to transform, in an appropriate manner, the equations to a new system, such that the interval of existence for the solutions of this new system is <span class="texhtml">[0,∞)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Singularities_of_the_n-body_problem">Singularities of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=13" title="Edit section: Singularities of the n-body problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There can be two types of singularities of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem: </p> <ul><li>collisions of two or more bodies, but for which <span class="texhtml"><b>q</b>(<i>t</i>)</span> (the bodies' positions) remains finite. (In this mathematical sense, a "collision" means that two pointlike bodies have identical positions in space.)</li> <li>singularities in which a collision does not occur, but <span class="texhtml"><b>q</b>(<i>t</i>)</span> does not remain finite. In this scenario, bodies diverge to infinity in a finite time, while at the same time tending towards zero separation (an imaginary collision occurs "at infinity").</li></ul> <p>The latter ones are called Painlevé's conjecture (no-collisions singularities). Their existence has been conjectured for <span class="texhtml"><i>n</i> > 3</span> by <a href="/wiki/Paul_Painlev%C3%A9" title="Paul Painlevé">Painlevé</a> (see <a href="/wiki/Painlev%C3%A9_conjecture" title="Painlevé conjecture">Painlevé conjecture</a>). Examples of this behavior for <span class="texhtml"><i>n</i> = 5</span> have been constructed by Xia<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> and a heuristic model for <span class="texhtml"><i>n</i> = 4</span> by Gerver.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Donald_G._Saari" title="Donald G. Saari">Donald G. Saari</a> has shown that for 4 or fewer bodies, the set of initial data giving rise to singularities has <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">measure</a> zero.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Simulation">Simulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=14" title="Edit section: Simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/N-body_simulation" title="N-body simulation">n-body simulation</a></div> <p>While there are analytic solutions available for the classical (i.e. nonrelativistic) two-body problem and for selected configurations with <span class="texhtml"><i>n</i> > 2</span>, in general <span class="texhtml mvar" style="font-style:italic;">n</span>-body problems must be solved or simulated using numerical methods.<sup id="cite_ref-Trenti_2008_21-1" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Few_bodies">Few bodies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=15" title="Edit section: Few bodies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a small number of bodies, an <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem can be solved using <i>direct methods</i>, also called <i>particle–particle methods</i>. These methods numerically integrate the differential equations of motion. Numerical integration for this problem can be a challenge for several reasons. First, the <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a> is singular; it goes to infinity as the distance between two particles goes to zero. The gravitational potential may be "softened" to remove the singularity at small distances:<sup id="cite_ref-Trenti_2008_21-2" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\varepsilon }=\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\sqrt {\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}+\varepsilon ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <msqrt> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\varepsilon }=\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\sqrt {\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}+\varepsilon ^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71efea3e6afca3a60f0a04bd515b3f7b944933c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:31.849ex; height:8.509ex;" alt="{\displaystyle U_{\varepsilon }=\sum _{1\leq i<j\leq n}{\frac {Gm_{i}m_{j}}{\sqrt {\left\|\mathbf {q} _{j}-\mathbf {q} _{i}\right\|^{2}+\varepsilon ^{2}}}}}"></span> Second, in general for <span class="texhtml"><i>n</i> > 2</span>, the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem is <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic</a>,<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> which means that even small errors in integration may grow exponentially in time. Third, a simulation may be over large stretches of model time (e.g. millions of years) and numerical errors accumulate as integration time increases. </p><p>There are a number of techniques to reduce errors in numerical integration.<sup id="cite_ref-Trenti_2008_21-3" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Local coordinate systems are used to deal with widely differing scales in some problems, for example an Earth–Moon coordinate system in the context of a solar system simulation. Variational methods and perturbation theory can yield approximate analytic trajectories upon which the numerical integration can be a correction. The use of a <a href="/wiki/Symplectic_integrator" title="Symplectic integrator">symplectic integrator</a> ensures that the simulation obeys Hamilton's equations to a high degree of accuracy and in particular that energy is conserved. </p> <div class="mw-heading mw-heading3"><h3 id="Many_bodies">Many bodies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=16" title="Edit section: Many bodies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Direct methods using numerical integration require on the order of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>n</i><sup>2</sup></span> computations to evaluate the potential energy over all pairs of particles, and thus have a <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> of <span class="texhtml"><i>O</i>(<i>n</i><sup>2</sup>)</span>. For simulations with many particles, the <span class="texhtml"><i>O</i>(<i>n</i><sup>2</sup>)</span> factor makes large-scale calculations especially time-consuming.<sup id="cite_ref-Trenti_2008_21-4" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>A number of approximate methods have been developed that reduce the time complexity relative to direct methods:<sup id="cite_ref-Trenti_2008_21-5" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <ul><li><b>Tree code methods</b>, such as a <a href="/wiki/Barnes%E2%80%93Hut_simulation" title="Barnes–Hut simulation">Barnes–Hut simulation</a>, are spatially-hierarchical methods used when distant particle contributions do not need to be computed to high accuracy. The potential of a distant group of particles is computed using a <a href="/wiki/Multipole_expansion" title="Multipole expansion">multipole expansion</a> or other approximation of the potential. This allows for a reduction in complexity to <span class="texhtml"><i>O</i>(<i>n</i> log <i>n</i>)</span>.</li> <li><b><a href="/wiki/Fast_multipole_method" title="Fast multipole method">Fast multipole methods</a></b> take advantage of the fact that the multipole-expanded forces from distant particles are similar for particles close to each other, and uses local expansions of far-field forces to reduce computational effort. It is claimed that this further approximation reduces the complexity to <span class="texhtml"><i>O</i>(<i>n</i>)</span>.<sup id="cite_ref-Trenti_2008_21-6" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/N-body_simulation#Particle_mesh_method" title="N-body simulation">Particle mesh methods</a></b> divide up simulation space into a three dimensional grid onto which the mass density of the particles is interpolated. Then calculating the potential becomes a matter of solving a <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a> on the grid, which can be computed in <span class="texhtml"><i>O</i>(<i>n</i> log <i>n</i>)</span> time using <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> or <span class="texhtml"><i>O</i>(<i>n</i>)</span> time using <a href="/wiki/Multigrid" class="mw-redirect" title="Multigrid">multigrid</a> techniques. This can provide fast solutions at the cost of higher error for short-range forces. <a href="/wiki/Adaptive_mesh_refinement" title="Adaptive mesh refinement">Adaptive mesh refinement</a> can be used to increase accuracy in regions with large numbers of particles.</li> <li><b><a href="/wiki/P3M" title="P3M">P<sup>3</sup>M</a></b> and <b>PM-tree methods</b> are hybrid methods that use the particle mesh approximation for distant particles, but use more accurate methods for close particles (within a few grid intervals). P<sup>3</sup>M stands for <i>particle–particle, particle–mesh</i> and uses direct methods with softened potentials at close range. PM-tree methods instead use tree codes at close range. As with particle mesh methods, adaptive meshes can increase computational efficiency.</li> <li><b><a href="/wiki/Mean_field" class="mw-redirect" title="Mean field">Mean field</a> methods</b> approximate the system of particles with a time-dependent <a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a> representing the mass density that is coupled to a self-consistent Poisson equation representing the potential. It is a type of <a href="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">smoothed-particle hydrodynamics</a> approximation suitable for large systems.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Strong_gravitation">Strong gravitation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=17" title="Edit section: Strong gravitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In astrophysical systems with strong gravitational fields, such as those near the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> of a <a href="/wiki/Black_hole" title="Black hole">black hole</a>, <span class="texhtml mvar" style="font-style:italic;">n</span>-body simulations must take into account <a href="/wiki/General_relativity" title="General relativity">general relativity</a>; such simulations are the domain of <a href="/wiki/Numerical_relativity" title="Numerical relativity">numerical relativity</a>. Numerically simulating the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> is extremely challenging<sup id="cite_ref-Trenti_2008_21-7" class="reference"><a href="#cite_note-Trenti_2008-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and a <a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized post-Newtonian formalism</a> (PPN), such as the <a href="/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations" title="Einstein–Infeld–Hoffmann equations">Einstein–Infeld–Hoffmann equations</a>, is used if possible. The <a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">two-body problem in general relativity</a> is analytically solvable only for the Kepler problem, in which one mass is assumed to be much larger than the other.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_n-body_problems">Other <span class="texhtml mvar" style="font-style:italic;">n</span>-body problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=18" title="Edit section: Other n-body problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most work done on the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem has been on the gravitational problem. But there exist other systems for which <span class="texhtml mvar" style="font-style:italic;">n</span>-body mathematics and simulation techniques have proven useful. </p><p>In large scale <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a> problems, such as the simulation of <a href="/wiki/Protein" title="Protein">proteins</a> and cellular assemblies in <a href="/wiki/Structural_biology" title="Structural biology">structural biology</a>, the <a href="/wiki/Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a> has the same form as the gravitational potential, except that charges may be positive or negative, leading to repulsive as well as attractive forces.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> <i>Fast Coulomb solvers</i> are the electrostatic counterpart to fast multipole method simulators. These are often used with <a href="/wiki/Periodic_boundary_conditions" title="Periodic boundary conditions">periodic boundary conditions</a> on the region simulated and <a href="/wiki/Ewald_summation" title="Ewald summation">Ewald summation</a> techniques are used to speed up computations.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, some models have <a href="/wiki/Loss_function" title="Loss function">loss functions</a> of a form similar to that of the gravitational potential: a sum of kernel functions over all pairs of objects, where the kernel function depends on the distance between the objects in parameter space.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Example problems that fit into this form include <a href="/wiki/Nearest_neighbor_search#All_nearest_neighbors" title="Nearest neighbor search">all-nearest-neighbors</a> in <a href="/wiki/Manifold_learning" class="mw-redirect" title="Manifold learning">manifold learning</a>, <a href="/wiki/Kernel_density_estimation" title="Kernel density estimation">kernel density estimation</a>, and <a href="/wiki/Kernel_machine" class="mw-redirect" title="Kernel machine">kernel machines</a>. Alternative optimizations to reduce the <span class="texhtml"><i>O</i>(<i>n</i><sup>2</sup>)</span> time complexity to <span class="texhtml"><i>O</i>(<i>n</i>)</span> have been developed, such as <i>dual tree</i> algorithms, that have applicability to the gravitational <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem as well. </p><p>A technique in <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">Computational fluid dynamics</a> called Vortex Methods sees the <a href="/wiki/Vorticity" title="Vorticity">vorticity</a> in a fluid domain discretized onto particles which are then advected with the velocity at their centers. Because the fluid velocity and vorticity are related via a <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a>, the velocity can be solved in the same manner as gravitation and electrostatics: as an <span class="texhtml mvar" style="font-style:italic;">n</span>-body summation over all vorticity-containing particles. The summation uses the <a href="/wiki/Biot-Savart_law" class="mw-redirect" title="Biot-Savart law">Biot-Savart law</a>, with vorticity taking the place of electrical current.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> In the context of particle-laden turbulent multiphase flows, determining an overall disturbance field generated by all particles is an <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem. If the particles translating within the flow are much smaller than the flow's Kolmogorov scale, their linear Stokes disturbance fields can be superposed, yielding a system of 3<span class="texhtml mvar" style="font-style:italic;">n</span> equations for 3 components of disturbance velocities at the location of <span class="texhtml mvar" style="font-style:italic;">n</span> particles.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial mechanics</a></li> <li><a href="/wiki/Gravitational_two-body_problem" class="mw-redirect" title="Gravitational two-body problem">Gravitational two-body problem</a></li> <li><a href="/wiki/Jacobi_integral" title="Jacobi integral">Jacobi integral</a></li> <li><a href="/wiki/Lunar_theory" title="Lunar theory">Lunar theory</a></li> <li><a href="/wiki/Natural_units" title="Natural units">Natural units</a></li> <li><a href="/wiki/Numerical_model_of_the_Solar_System" title="Numerical model of the Solar System">Numerical model of the Solar System</a></li> <li><a href="/wiki/Stability_of_the_Solar_System" title="Stability of the Solar System">Stability of the Solar System</a></li> <li><a href="/wiki/Few-body_systems" title="Few-body systems">Few-body systems</a></li> <li><a href="/wiki/N-body_simulation" title="N-body simulation">N-body simulation</a>, a method for numerically obtaining trajectories of bodies in an N-body system.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=20" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Leimanis_and_Minorsky-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Leimanis_and_Minorsky_1-0">^</a></b></span> <span class="reference-text">Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized <span class="texhtml mvar" style="font-style:italic;">n</span>-body Problem").</span> </li> <li id="cite_note-Heggie_and_Hut-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Heggie_and_Hut_2-0">^</a></b></span> <span class="reference-text">See references cited for Heggie and Hut.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><i>Quasi-steady</i> loads are the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, under a <i>steady-state</i> condition, a system's state is invariant to time; otherwise, the first derivatives and all higher derivatives are zero.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">R. M. Rosenberg states the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem similarly (see References): "Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?" Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary <span class="texhtml mvar" style="font-style:italic;">n</span> can be approximated via <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, but in practice such an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem may be solved using <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a>, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book <i>Gravitational <span class="texhtml mvar" style="font-style:italic;">n</span>-Body Simulations</i> listed in the References.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClarkClark2001" class="citation book cs1">Clark, David H.; Clark, Stephen P. H. (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/newtonstyrannysu00clar"><i>The Suppressed Scientific Discoveries of Stephen Gray and John Flamsteed, Newton's Tyranny</i></a></span>. W. H. Freeman and Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Suppressed+Scientific+Discoveries+of+Stephen+Gray+and+John+Flamsteed%2C+Newton%27s+Tyranny&rft.pub=W.+H.+Freeman+and+Co.&rft.date=2001&rft.aulast=Clark&rft.aufirst=David+H.&rft.au=Clark%2C+Stephen+P.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnewtonstyrannysu00clar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span>. A popularization of the historical events and bickering between those parties, but more importantly about the results they produced.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrewster1905" class="citation book cs1"><a href="/wiki/Sir_David_Brewster" class="mw-redirect" title="Sir David Brewster">Brewster, David</a> (1905). "Discovery of gravitation, A.D. 1666". In Johnson, Rossiter (ed.). <i>The Great Events by Famous Historians</i>. Vol. XII. The National Alumni. pp. 51–65.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Discovery+of+gravitation%2C+A.D.+1666&rft.btitle=The+Great+Events+by+Famous+Historians&rft.pages=51-65&rft.pub=The+National+Alumni&rft.date=1905&rft.aulast=Brewster&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Rudolf Kurth has an extensive discussion in his book (see References) on planetary perturbations. An aside: these mathematically undefined planetary perturbations (wobbles) still exist undefined even today and planetary orbits have to be constantly updated, usually yearly. See Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, prepared jointly by the Nautical Almanac Offices of the United Kingdom and the United States of America.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">See <i>Principia</i>, Book Three, <i>System of the World</i>, "General Scholium", page 372, last paragraph. Newton was well aware that his mathematical model did not reflect physical reality. This edition referenced is from the <i>Great Books of the Western World</i>, Volume 34, which was translated by Andrew Motte and revised by <a href="/wiki/Florian_Cajori" title="Florian Cajori">Florian Cajori</a>.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup> This same paragraph is on page 1160 in <a href="/wiki/Stephen_Hawkins" title="Stephen Hawkins">Stephen Hawkins</a>, <i>On the Shoulders of Giants</i>, 2002 edition;<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup> is a copy from Daniel Adee's 1848 addition. Cohen also has translated new editions: <i>Introduction to Newton's Principia</i>, 1970; and <i>Isaac Newton's Principia, with Variant Readings</i>, 1972. Cajori also wrote <i>History of Science</i>, which is online.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-Cohen-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cohen_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cohen_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See I. Bernard Cohen's <i>Scientific American</i> article.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">For details of the serious error in Poincare's first submission see the article by Diacu.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBabadzanjanz1979" class="citation cs2">Babadzanjanz, L. K. (1979), "Existence of the continuations in the <i>N</i>-body problem", <i>Celestial Mechanics</i>, <b>20</b> (1): 43–57, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979CeMec..20...43B">1979CeMec..20...43B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01236607">10.1007/BF01236607</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0538663">0538663</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120358878">120358878</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics&rft.atitle=Existence+of+the+continuations+in+the+N-body+problem&rft.volume=20&rft.issue=1&rft.pages=43-57&rft.date=1979&rft_id=info%3Adoi%2F10.1007%2FBF01236607&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D538663%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120358878%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1979CeMec..20...43B&rft.aulast=Babadzanjanz&rft.aufirst=L.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBabadzanjanz1993" class="citation cs2">Babadzanjanz, L. K. (1993), "On the global solution of the <i>N</i>-body problem", <i>Celestial Mechanics and Dynamical Astronomy</i>, <b>56</b> (3): 427–449, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993CeMDA..56..427B">1993CeMDA..56..427B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00691812">10.1007/BF00691812</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1225892">1225892</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120617936">120617936</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics+and+Dynamical+Astronomy&rft.atitle=On+the+global+solution+of+the+N-body+problem&rft.volume=56&rft.issue=3&rft.pages=427-449&rft.date=1993&rft_id=info%3Adoi%2F10.1007%2FBF00691812&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1225892%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120617936%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1993CeMDA..56..427B&rft.aulast=Babadzanjanz&rft.aufirst=L.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang1991" class="citation cs2"><a href="/wiki/Qiudong_Wang" title="Qiudong Wang">Wang, Qiu Dong</a> (1991), "The global solution of the <i>n</i>-body problem", <i>Celestial Mechanics and Dynamical Astronomy</i>, <b>50</b> (1): 73–88, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991CeMDA..50...73W">1991CeMDA..50...73W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00048987">10.1007/BF00048987</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1117788">1117788</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118132097">118132097</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics+and+Dynamical+Astronomy&rft.atitle=The+global+solution+of+the+n-body+problem&rft.volume=50&rft.issue=1&rft.pages=73-88&rft.date=1991&rft_id=info%3Adoi%2F10.1007%2FBF00048987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1117788%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118132097%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1991CeMDA..50...73W&rft.aulast=Wang&rft.aufirst=Qiu+Dong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Meyer 2009, pp. 27–28</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Meyer 2009, p. 28</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Meyer 2009, pp. 28–29</span> </li> <li id="cite_note-Chenciner_2007-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chenciner_2007_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chenciner_2007_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Chenciner_2007_18-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Chenciner 2007</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Meyer 2009, p. 34</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.uio.no/studier/emner/matnat/astro/AST1100/h09/undervisningsmateriale/lecture5.pdf">"AST1100 Lecture Notes: 5 The virial theorem"</a> <span class="cs1-format">(PDF)</span>. University of Oslo<span class="reference-accessdate">. Retrieved <span class="nowrap">25 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=AST1100+Lecture+Notes%3A+5+The+virial+theorem&rft.pub=University+of+Oslo&rft_id=http%3A%2F%2Fwww.uio.no%2Fstudier%2Femner%2Fmatnat%2Fastro%2FAST1100%2Fh09%2Fundervisningsmateriale%2Flecture5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-Trenti_2008-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Trenti_2008_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Trenti_2008_21-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text">Trenti 2008</span> </li> <li id="cite_note-Bate,_Mueller,_and_White-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bate,_Mueller,_and_White_22-0">^</a></b></span> <span class="reference-text">See Bate, Mueller, and White, Chapter 1: "Two-Body Orbital Mechanics", pp. 1–49. These authors were from the Department of Astronautics and Computer Science, United States Air Force Academy. Their textbook is not filled with advanced mathematics.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">For the classical approach, if the common <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> (i.e., the barycenter) of the two bodies is considered to be at rest, then each body travels along a <a href="/wiki/Conic_section" title="Conic section">conic section</a> which has a <a href="/wiki/Focus_(geometry)" title="Focus (geometry)">focus</a> at the barycenter of the system. In the case of a hyperbola it has the branch at the side of that focus. The two conics will be in the same plane. The type of conic (<a href="/wiki/Circle" title="Circle">circle</a>, <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, <a href="/wiki/Parabola" title="Parabola">parabola</a> or <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>) is determined by finding the sum of the combined kinetic energy of two bodies and the <a href="/wiki/Potential_energy#Gravitational_potential_energy" title="Potential energy">potential energy</a> when the bodies are far apart. (This potential energy is always a negative value; energy of rotation of the bodies about their axes is not counted here) <ul><li>If the sum of the energies is negative, then they both trace out ellipses.</li> <li>If the sum of both energies is zero, then they both trace out parabolas. As the distance between the bodies tends to infinity, their relative speed tends to zero.</li> <li>If the sum of both energies is positive, then they both trace out hyperbolas. As the distance between the bodies tends to infinity, their relative speed tends to some positive number.</li></ul> </span></li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">For this approach see Lindsay's <i>Physical Mechanics</i>, Chapter 3: "Curvilinear Motion in a Plane", and specifically paragraphs 3–9, "Planetary Motion"; pp. 83–96. Lindsay presentation goes a long way in explaining these latter comments for the fixed two-body problem; i.e., when the Sun is assumed fixed.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Note: The fact a parabolic orbit has zero energy arises from the assumption the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy by convention.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Science Program's <i>The Nature of the Universe</i> states Clarence Cleminshaw (1902–1985) served as assistant director of <i>Griffith Observatory</i> from 1938 to 1958 and as director from 1958 to 1969. Some publications by Cleminshaw: <ul><li>Cleminshaw, C. H.: "Celestial Speeds", 4 1953, equation, Kepler, orbit, comet, Saturn, Mars, velocity.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup></li> <li>Cleminshaw, C. H.: "The Coming Conjunction of Jupiter and Saturn", 7 1960, Saturn, Jupiter, observe, conjunction.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup></li> <li>Cleminshaw, C. H.: "The Scale of The Solar System", 7 1959, Solar system, scale, Jupiter, sun, size, light.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (March 2017)">full citation needed</span></a></i>]</sup></li></ul> </span></li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrush1983" class="citation book cs1">Brush, Stephen G., ed. (1983). <i>Maxwell on Saturn's Rings</i>. MIT Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maxwell+on+Saturn%27s+Rings&rft.pub=MIT+Press&rft.date=1983&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">See Leimanis and Minorsky's historical comments.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">See Moulton's <i>Restricted Three-body Problem</i> for its analytical and graphical solution.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">See Meirovitch's book: Chapters 11: "Problems in Celestial Mechanics"; 12; "Problem in Spacecraft Dynamics"; and Appendix A: "Dyadics".</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuang1960" class="citation journal cs1">Huang, Su-Shu (1960). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=smhIAQAAIAAJ&q=huang+very+restricted&pg=PA354">"Very Restricted Four-Body Problem"</a>. <i>NASA TND-501</i>. <b>65</b>: 347. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960AJ.....65S.347H">1960AJ.....65S.347H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F108151">10.1086/108151</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F19890068606">2060/19890068606</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NASA+TND-501&rft.atitle=Very+Restricted+Four-Body+Problem&rft.volume=65&rft.pages=347&rft.date=1960&rft_id=info%3Ahdl%2F2060%2F19890068606&rft_id=info%3Adoi%2F10.1086%2F108151&rft_id=info%3Abibcode%2F1960AJ.....65S.347H&rft.aulast=Huang&rft.aufirst=Su-Shu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsmhIAQAAIAAJ%26q%3Dhuang%2Bvery%2Brestricted%26pg%3DPA354&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNegriPrado2020" class="citation journal cs1">Negri, Rodolfo B.; Prado, Antonio F. B. A. (2020). "Generalizing the Bicircular Restricted Four-Body Problem". <i>Journal of Guidance, Control, and Dynamics</i>. <b>43</b> (6): 1173–1179. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020JGCD...43.1173N">2020JGCD...43.1173N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F1.G004848">10.2514/1.G004848</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:213600592">213600592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Guidance%2C+Control%2C+and+Dynamics&rft.atitle=Generalizing+the+Bicircular+Restricted+Four-Body+Problem&rft.volume=43&rft.issue=6&rft.pages=1173-1179&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A213600592%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2514%2F1.G004848&rft_id=info%3Abibcode%2F2020JGCD...43.1173N&rft.aulast=Negri&rft.aufirst=Rodolfo+B.&rft.au=Prado%2C+Antonio+F.+B.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-Chierchia_2010-33"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chierchia_2010_33-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chierchia_2010_33-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Chierchia_2010_33-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Chierchia 2010</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Féjoz 2004</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">See Chierchia 2010 for animations illustrating homographic motions.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Celletti 2008</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoore1993" class="citation journal cs1">Moore, Cristopher (1993-06-14). 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"A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses". <i>The Annals of Mathematics</i>. <b>152</b> (3): 881. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0011268">math/0011268</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000math.....11268C">2000math.....11268C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2661357">10.2307/2661357</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2661357">2661357</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10024592">10024592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematics&rft.atitle=A+Remarkable+Periodic+Solution+of+the+Three-Body+Problem+in+the+Case+of+Equal+Masses&rft.volume=152&rft.issue=3&rft.pages=881&rft.date=2000-11&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2661357%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F2000math.....11268C&rft_id=info%3Aarxiv%2Fmath%2F0011268&rft_id=info%3Adoi%2F10.2307%2F2661357&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10024592%23id-name%3DS2CID&rft.aulast=Chenciner&rft.aufirst=Alain&rft.au=Montgomery%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQiu-Dong1990" class="citation journal cs1">Qiu-Dong, Wang (1990-03-01). 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Princeton: D. 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Paris: Gauthier-Villars Et Fils: 27. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1908%2F4228">1908/4228</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/951409281">951409281</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Lilliad+-+Universit%C3%A9+de+Lille+-+Sciences+et+Technologies&rft.atitle=Trait%C3%A9+de+M%C3%A9canique+C%C3%A9leste&rft.volume=III&rft.pages=27&rft.date=1894&rft_id=info%3Ahdl%2F1908%2F4228&rft_id=info%3Aoclcnum%2F951409281&rft.aulast=Tisserand&rft.aufirst=Fran%C3%A7ois+F%C3%A9lix&rft_id=https%3A%2F%2Firis.univ-lille.fr%2Fbitstream%2Fhandle%2F1908%2F4228%2FQ11840-3.pdf%3Fsequence%3D4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrentiHut2008" class="citation journal cs1">Trenti, Michele; Hut, Piet (2008). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.3930">"<span class="texhtml mvar" style="font-style:italic;">n</span>-body simulations"</a>. <i>Scholarpedia</i>. <b>3</b> (5): 3930. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SchpJ...3.3930T">2008SchpJ...3.3930T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.3930">10.4249/scholarpedia.3930</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-body+simulations&rft.volume=3&rft.issue=5&rft.pages=3930&rft.date=2008&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.3930&rft_id=info%3Abibcode%2F2008SchpJ...3.3930T&rft.aulast=Trenti&rft.aufirst=Michele&rft.au=Hut%2C+Piet&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.3930&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTruesdell1968" class="citation book cs1">Truesdell, Clifford (1968). <i>Essays in the History of Mechanics</i>. Berlin; Heidelberg: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-86649-4" title="Special:BookSources/978-3-642-86649-4"><bdi>978-3-642-86649-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+in+the+History+of+Mechanics&rft.place=Berlin%3B+Heidelberg&rft.pub=Springer-Verlag&rft.date=1968&rft.isbn=978-3-642-86649-4&rft.aulast=Truesdell&rft.aufirst=Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Winter1970" class="citation book cs1">Van Winter, Clasine (1970). "The <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem on a Hilbert space of analytic functions". In <a href="/wiki/Roger_G._Newton" title="Roger G. Newton">Gilbert, Robert P.</a>; Newton, Roger G. (eds.). <i>Analytic Methods in Mathematical Physics</i>. New York: Gordon and Breach. pp. 569–578. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/848738761">848738761</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-body+problem+on+a+Hilbert+space+of+analytic+functions&rft.btitle=Analytic+Methods+in+Mathematical+Physics&rft.place=New+York&rft.pages=569-578&rft.pub=Gordon+and+Breach&rft.date=1970&rft_id=info%3Aoclcnum%2F848738761&rft.aulast=Van+Winter&rft.aufirst=Clasine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang1991" class="citation journal cs1"><a href="/wiki/Qiudong_Wang" title="Qiudong Wang">Wang, Qiudong</a> (1991). "The global solution of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem". <i><a href="/wiki/Celestial_Mechanics_and_Dynamical_Astronomy" title="Celestial Mechanics and Dynamical Astronomy">Celestial Mechanics and Dynamical Astronomy</a></i>. <b>50</b> (1): 73–88. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991CeMDA..50...73W">1991CeMDA..50...73W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00048987">10.1007/BF00048987</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0923-2958">0923-2958</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1117788">1117788</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118132097">118132097</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics+and+Dynamical+Astronomy&rft.atitle=The+global+solution+of+the+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-body+problem&rft.volume=50&rft.issue=1&rft.pages=73-88&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118132097%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1991CeMDA..50...73W&rft.issn=0923-2958&rft_id=info%3Adoi%2F10.1007%2FBF00048987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1117788%23id-name%3DMR&rft.aulast=Wang&rft.aufirst=Qiudong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXia1992" class="citation journal cs1">Xia, Zhihong (1992). "The Existence of Noncollision Singularities in Newtonian Systems". <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. <b>135</b> (3): 411–468. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2946572">10.2307/2946572</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2946572">2946572</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=The+Existence+of+Noncollision+Singularities+in+Newtonian+Systems&rft.volume=135&rft.issue=3&rft.pages=411-468&rft.date=1992&rft_id=info%3Adoi%2F10.2307%2F2946572&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2946572%23id-name%3DJSTOR&rft.aulast=Xia&rft.aufirst=Zhihong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=22" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Further_reading_cleanup plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This "<a href="/wiki/Wikipedia:Manual_of_Style/Layout#Further_reading" title="Wikipedia:Manual of Style/Layout">Further reading</a>" section <b>may need cleanup</b>.<span class="hide-when-compact"> Please read the <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">editing guide</a> and help improve the section.</span> <span class="date-container"><i>(<span class="date">March 2017</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBattin1987" class="citation book cs1">Battin, Richard H. (1987). <i>An Introduction to The Mathematics and Methods of Astrodynamics</i>. AIAA.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+The+Mathematics+and+Methods+of+Astrodynamics&rft.pub=AIAA&rft.date=1987&rft.aulast=Battin&rft.aufirst=Richard+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span> Employs energy methods rather than a Newtonian approach.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoccalettiPucacco1998" class="citation book cs1">Boccaletti, D.; Pucacco, G. (1998). <i>Theory of Orbits</i>. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Orbits&rft.pub=Springer-Verlag&rft.date=1998&rft.aulast=Boccaletti&rft.aufirst=D.&rft.au=Pucacco%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrouwerClemence1961" class="citation book cs1">Brouwer, Dirk; Clemence, Gerald M. (1961). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/methodsofcelesti00brou"><i>Methods of Celestial Mechanics</i></a></span>. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Celestial+Mechanics&rft.pub=Academic+Press&rft.date=1961&rft.aulast=Brouwer&rft.aufirst=Dirk&rft.au=Clemence%2C+Gerald+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmethodsofcelesti00brou&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrandall1996" class="citation book cs1">Crandall, Richard E. (1996). "Chapter 5: "Nonlinear & Complex Systems"; paragraph 5.1: "<span class="texhtml mvar" style="font-style:italic;">n</span>-body problems & chaos"<span class="cs1-kern-right"></span>". <i>Topics in Advanced Scientific Computation</i>. 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"Applied Matrix and Tensor Analysis". <i>Physics Today</i>. <b>25</b> (12): 55. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1972PhT....25l..55E">1972PhT....25l..55E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3071146">10.1063/1.3071146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Applied+Matrix+and+Tensor+Analysis&rft.volume=25&rft.issue=12&rft.pages=55&rft.date=1970&rft_id=info%3Adoi%2F10.1063%2F1.3071146&rft_id=info%3Abibcode%2F1972PhT....25l..55E&rft.aulast=Eisele&rft.aufirst=John+A.&rft.au=Mason%2C+Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelman1968" class="citation journal cs1">Gelman, Harry (1968). 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Sci.)</i>. <b>1968</b> (3).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Res.+NBS+72B+%28Math.+Sci.%29&rft.atitle=The+intrinsic+vector&rft.volume=1968&rft.issue=3&rft.date=1968&rft.aulast=Gelman&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span><br /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelman1969" class="citation journal cs1">Gelman, Harry (1969). "The Conjugacy Theorem". <i>J. Res. NBS 72B (Math. Sci.)</i>. <b>1969</b> (2).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Res.+NBS+72B+%28Math.+Sci.%29&rft.atitle=The+Conjugacy+Theorem&rft.volume=1969&rft.issue=2&rft.date=1969&rft.aulast=Gelman&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span><br /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelman1971" class="citation journal cs1">Gelman, Harry (October 1971). "A Note on the time dependence of the effective axis and angle of a rotation". <i>J. Res. NBS 72B (Math. Sci.)</i>. <b>1971</b> (3–4).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Res.+NBS+72B+%28Math.+Sci.%29&rft.atitle=A+Note+on+the+time+dependence+of+the+effective+axis+and+angle+of+a+rotation&rft.volume=1971&rft.issue=3%E2%80%934&rft.date=1971-10&rft.aulast=Gelman&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHagihara1970" class="citation book cs1 cs1-prop-long-vol">Hagihara, Y. (1970). <i>Celestial Mechanics</i>. Vol. I, II pt 1, II pt 2. MIT Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Celestial+Mechanics&rft.pub=MIT+Press&rft.date=1970&rft.aulast=Hagihara&rft.aufirst=Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKorenev1967" class="citation book cs1">Korenev, G. V. (1967). <i>The Mechanics of Guided Bodies</i>. CRC Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mechanics+of+Guided+Bodies&rft.pub=CRC+Press&rft.date=1967&rft.aulast=Korenev&rft.aufirst=G.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeriam1978" class="citation book cs1">Meriam, J. L. (1978). <i>Engineering Mechanics</i>. Vol. 1–2. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pub=John+Wiley+%26+Sons&rft.date=1978&rft.aulast=Meriam&rft.aufirst=J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurrayDermott2000" class="citation book cs1">Murray, Carl D.; Dermott, Stanley F. (2000). <i>Solar System Dynamics</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solar+System+Dynamics&rft.pub=Cambridge+University+Press&rft.date=2000&rft.aulast=Murray&rft.aufirst=Carl+D.&rft.au=Dermott%2C+Stanley+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuadling1994" class="citation book cs1">Quadling, Henley (June 1994). <i>Gravitational <span class="texhtml mvar" style="font-style:italic;">n</span>-Body Simulation: 16 bit DOS version</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitational+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-Body+Simulation%3A+16+bit+DOS+version&rft.date=1994-06&rft.aulast=Quadling&rft.aufirst=Henley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span> nbody*.zip is available at <a rel="nofollow" class="external free" href="https://web.archive.org/web/19990221123102/http://ftp.cica.indiana.edu/">https://web.archive.org/web/19990221123102/http://ftp.cica.indiana.edu/</a>: see external links.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaari1990" class="citation journal cs1">Saari, D. (1990). "A visit to the Newtonian <span class="texhtml mvar" style="font-style:italic;">n</span>-body problem via Elementary Complex Variables". <i>American Mathematical Monthly</i>. <b>89</b> (2): 105–119. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2323910">10.2307/2323910</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2323910">2323910</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+visit+to+the+Newtonian+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-body+problem+via+Elementary+Complex+Variables&rft.volume=89&rft.issue=2&rft.pages=105-119&rft.date=1990&rft_id=info%3Adoi%2F10.2307%2F2323910&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2323910%23id-name%3DJSTOR&rft.aulast=Saari&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaariHulkower1981" class="citation journal cs1">Saari, D. G.; Hulkower, N. D. (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0396%2881%2990051-6">"On the Manifolds of Total Collapse Orbits and of Completely Parabolic Orbits for the <span class="texhtml mvar" style="font-style:italic;">n</span>-Body Problem"</a>. <i>Journal of Differential Equations</i>. <b>41</b> (1): 27–43. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981JDE....41...27S">1981JDE....41...27S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0396%2881%2990051-6">10.1016/0022-0396(81)90051-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Differential+Equations&rft.atitle=On+the+Manifolds+of+Total+Collapse+Orbits+and+of+Completely+Parabolic+Orbits+for+the+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-Body+Problem&rft.volume=41&rft.issue=1&rft.pages=27-43&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0022-0396%2881%2990051-6&rft_id=info%3Abibcode%2F1981JDE....41...27S&rft.aulast=Saari&rft.aufirst=D.+G.&rft.au=Hulkower%2C+N.+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0022-0396%252881%252990051-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzebehely1967" class="citation book cs1">Szebehely, Victor (1967). <i>Theory of Orbits</i>. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Orbits&rft.pub=Academic+Press&rft.date=1967&rft.aulast=Szebehely&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AN-body+problem" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N-body_problem&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Three_Body_Problem">Three-Body Problem</a> at <a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20021017171212/http://www.geom.umn.edu/~megraw/CR3BP_html/cr3bp.html">More detailed information on the three-body problem</a></li> <li><a rel="nofollow" class="external text" href="http://www.ifmo.ru/butikov/ManyBody.pdf">Regular Keplerian motions in classical many-body systems</a></li> <li><a rel="nofollow" class="external text" href="http://alecjacobson.com/programs/three-body-chaos/">Applet demonstrating chaos in restricted three-body problem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091017075952/http://alecjacobson.com/programs/three-body-chaos/">Archived</a> 2009-10-17 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.ifmo.ru/butikov/Projects/Collection.html">Applets demonstrating many different three-body motions</a></li> <li><a rel="nofollow" class="external text" href="http://www.datasync.com/~rsf1/manybod1.htm">On the integration of the <span class="texhtml mvar" style="font-style:italic;">n</span>-body equations</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161030011822/http://www.datasync.com/~rsf1/manybod1.htm">Archived</a> 2016-10-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.princeton.edu/~rvdb/JAVA/astro/galaxy/SolarSystem.html">Java applet simulating Solar System</a></li> <li><a rel="nofollow" class="external text" href="http://www.princeton.edu/~rvdb/JAVA/astro/galaxy/ThreeBody2.html">Java applet simulating a stable solution to the equi-mass 3-body problem</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090108060338/http://www.telefonica.net/web2/canrosin/index.htm">A java applet to simulate the 3D movement of set of particles under gravitational interaction</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090630215515/http://orinetz.com/planet/viewsysblog.php?specific=QUQTS2CSDQ44FDURR3XD6NUD6">Javascript Simulation of our Solar System</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051109070350/http://www.merlyn.demon.co.uk/gravity4.htm">The Lagrange Points</a> – with links to the original papers of Euler and Lagrange, and to translations, with discussion</li> <li><a rel="nofollow" class="external autonumber" href="http://ftp.math.utah.edu/pub/tex/bib/toc/sciam1980.html#242%281%29:January:1980">[1]</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/drons/nbody">Parallel GPU N-body simulation program with fast stackless particles tree traversal</a></li></ul> </div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 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dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Gravitational_orbits" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Orbits" title="Template:Orbits"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Orbits" title="Template talk:Orbits"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Orbits" title="Special:EditPage/Template:Orbits"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Gravitational_orbits" style="font-size:114%;margin:0 4em">Gravitational <a href="/wiki/Orbit" title="Orbit">orbits</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_orbits" title="List of orbits">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Box_orbit" title="Box orbit">Box</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Capture</a></li> <li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptical</a> / <a href="/wiki/Highly_elliptical_orbit" title="Highly elliptical orbit">Highly elliptical</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Escape</a></li> <li><a href="/wiki/Horseshoe_orbit" title="Horseshoe orbit">Horseshoe</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic trajectory</a></li> <li><a href="/wiki/Inclined_orbit" title="Inclined orbit">Inclined</a> / <a href="/wiki/Non-inclined_orbit" class="mw-redirect" title="Non-inclined orbit">Non-inclined</a></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange point</a></li> <li><a href="/wiki/Osculating_orbit" title="Osculating orbit">Osculating</a></li> <li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic trajectory</a></li> <li><a href="/wiki/Parking_orbit" title="Parking orbit">Parking</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Prograde / Retrograde</a></li> <li><a href="/wiki/Synchronous_orbit" title="Synchronous orbit">Synchronous</a> <ul><li><a href="/wiki/Semi-synchronous_orbit" title="Semi-synchronous orbit">semi</a></li> <li><a href="/wiki/Subsynchronous_orbit" title="Subsynchronous orbit">sub</a></li></ul></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Transfer orbit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em"><a href="/wiki/Geocentric_orbit" title="Geocentric orbit">Geocentric</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geosynchronous_orbit" title="Geosynchronous orbit">Geosynchronous</a> <ul><li><a href="/wiki/Geostationary_orbit" title="Geostationary orbit">Geostationary</a></li> <li><a href="/wiki/Geostationary_transfer_orbit" title="Geostationary transfer orbit">Geostationary transfer</a></li></ul></li> <li><a href="/wiki/Graveyard_orbit" title="Graveyard orbit">Graveyard</a></li> <li><a href="/wiki/High_Earth_orbit" title="High Earth orbit">High Earth</a></li> <li><a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">Low Earth</a></li> <li><a href="/wiki/Medium_Earth_orbit" title="Medium Earth orbit">Medium Earth</a></li> <li><a href="/wiki/Molniya_orbit" title="Molniya orbit">Molniya</a></li> <li><a href="/wiki/Near-equatorial_orbit" title="Near-equatorial orbit">Near-equatorial</a></li> <li><a href="/wiki/Orbit_of_the_Moon" title="Orbit of the Moon">Orbit of the Moon</a></li> <li><a href="/wiki/Polar_orbit" title="Polar orbit">Polar</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Sun-synchronous</a></li> <li><a href="/wiki/Transatmospheric_orbit" title="Transatmospheric orbit">Transatmospheric</a></li> <li><a href="/wiki/Tundra_orbit" title="Tundra orbit">Tundra</a></li> <li><a href="/wiki/Very_low_Earth_orbit" title="Very low Earth orbit">Very low Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">About<br />other points</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Mars <ul><li><a href="/wiki/Areocentric_orbit" title="Areocentric orbit">Areocentric</a></li> <li><a href="/wiki/Areosynchronous_orbit" title="Areosynchronous orbit">Areosynchronous</a></li> <li><a href="/wiki/Areostationary_orbit" title="Areostationary orbit">Areostationary</a></li></ul></li> <li>Lagrange points <ul><li><a href="/wiki/Distant_retrograde_orbit" title="Distant retrograde orbit">Distant retrograde</a></li> <li><a href="/wiki/Halo_orbit" title="Halo orbit">Halo</a></li> <li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous</a></li> <li><a href="/wiki/Libration_point_orbit" title="Libration point orbit">Libration</a></li></ul></li> <li><a href="/wiki/Lunar_orbit" title="Lunar orbit">Lunar</a></li> <li>Sun <ul><li><a href="/wiki/Heliocentric_orbit" title="Heliocentric orbit">Heliocentric</a> <ul><li><a href="/wiki/Earth%27s_orbit" title="Earth's orbit">Earth's orbit</a></li></ul></li> <li><a href="/wiki/Mars_cycler" title="Mars cycler">Mars cycler</a></li> <li><a href="/wiki/Sun-synchronous_orbit" title="Sun-synchronous orbit">Heliosynchronous</a></li></ul></li> <li>Other <ul><li><a href="/wiki/Lunar_cycler" title="Lunar cycler">Lunar cycler</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_elements" title="Orbital elements">Parameters</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em"><div class="hlist"><ul><li>Shape</li><li>Size</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">e</span>  <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><span class="texhtml mvar" style="font-style:italic;">a</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>  <a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-minor axis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Q</span>, <span class="texhtml mvar" style="font-style:italic;">q</span>  <a href="/wiki/Apsis" title="Apsis">Apsides</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Orientation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">i</span>  <a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><span class="texhtml mvar" style="font-style:italic;">Ω</span>  <a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">Longitude of the ascending node</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ω</span>  <a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ϖ</span>  <a href="/wiki/Longitude_of_the_periapsis" class="mw-redirect" title="Longitude of the periapsis">Longitude of the periapsis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Position</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">M</span>  <a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ν</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span>, <span class="texhtml mvar" style="font-style:italic;">f</span>  <a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">E</span>  <a href="/wiki/Eccentric_anomaly" title="Eccentric anomaly">Eccentric anomaly</a></li> <li><span class="texhtml mvar" style="font-style:italic;">L</span>  <a href="/wiki/Mean_longitude" title="Mean longitude">Mean longitude</a></li> <li><span class="texhtml mvar" style="font-style:italic;">l</span>  <a href="/wiki/True_longitude" title="True longitude">True longitude</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em">Variation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><span class="texhtml mvar" style="font-style:italic;">T</span>  <a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><span class="texhtml mvar" style="font-style:italic;">n</span>  <a href="/wiki/Mean_motion" title="Mean motion">Mean motion</a></li> <li><span class="texhtml mvar" style="font-style:italic;">v</span>  <a href="/wiki/Orbital_speed" title="Orbital speed">Orbital speed</a></li> <li><span class="texhtml"><i>t</i><sub>0</sub></span>  <a href="/wiki/Epoch_(astronomy)" title="Epoch (astronomy)">Epoch</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Maneuvers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer</a></li> <li><a href="/wiki/Collision_avoidance_(spacecraft)" title="Collision avoidance (spacecraft)">Collision avoidance (spacecraft)</a></li> <li><a href="/wiki/Delta-v" title="Delta-v">Delta-v</a></li> <li><a href="/wiki/Delta-v_budget" title="Delta-v budget">Delta-v budget</a></li> <li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Gravity_turn" title="Gravity turn">Gravity turn</a></li> <li><a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer</a></li> <li><a href="/wiki/Orbital_inclination_change" title="Orbital inclination change">Inclination change</a></li> <li><a href="/wiki/Low-energy_transfer" title="Low-energy transfer">Low-energy transfer</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li> <li><a href="/wiki/Orbit_phasing" title="Orbit phasing">Phasing</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Rocket equation</a></li> <li><a href="/wiki/Space_rendezvous" title="Space rendezvous">Rendezvous</a></li> <li><a href="/wiki/Trans-lunar_injection" title="Trans-lunar injection">Trans-lunar injection</a></li> <li><a href="/wiki/Transposition,_docking,_and_extraction" title="Transposition, docking, and extraction">Transposition, docking, and extraction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital<br />mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_coordinate_systems" title="Astronomical coordinate systems">Astronomical coordinate systems</a></li> <li><a href="/wiki/Characteristic_energy" title="Characteristic energy">Characteristic energy</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a></li> <li><a href="/wiki/Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial coordinate system</a></li> <li><a href="/wiki/Ground_track" class="mw-redirect" title="Ground track">Ground track</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Interplanetary_Transport_Network" title="Interplanetary Transport Network">Interplanetary Transport Network</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian point</a></li> <li><a class="mw-selflink selflink"><i>n</i>-body problem</a></li> <li><a href="/wiki/Orbit_equation" title="Orbit equation">Orbit equation</a></li> <li><a href="/wiki/Orbital_state_vectors" title="Orbital state vectors">Orbital state vectors</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbation</a></li> <li><a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">Retrograde and prograde motion</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">Specific angular momentum</a></li> <li><a href="/wiki/Two-line_element_set" title="Two-line element set">Two-line elements</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_orbits" title="List of orbits">List of orbits</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Numerical_methods_for_ordinary_differential_equations" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Numerical_integrators" title="Template:Numerical integrators"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numerical_integrators" title="Template talk:Numerical integrators"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numerical_integrators" title="Special:EditPage/Template:Numerical integrators"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Numerical_methods_for_ordinary_differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">First-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Backward_Euler_method" title="Backward Euler method">Backward Euler</a></li> <li><a href="/wiki/Semi-implicit_Euler_method" title="Semi-implicit Euler method">Semi-implicit Euler</a></li> <li><a href="/wiki/Exponential_Euler_method" class="mw-redirect" title="Exponential Euler method">Exponential Euler</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">Second-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Verlet_integration" title="Verlet integration">Verlet integration</a></li> <li><a href="/wiki/Velocity_Verlet" class="mw-redirect" title="Velocity Verlet">Velocity Verlet</a></li> <li><a href="/wiki/Trapezoidal_rule_(differential_equations)" title="Trapezoidal rule (differential equations)">Trapezoidal rule</a></li> <li><a href="/wiki/Beeman%27s_algorithm" title="Beeman's algorithm">Beeman's algorithm</a></li> <li><a href="/wiki/Midpoint_method" title="Midpoint method">Midpoint method</a></li> <li><a href="/wiki/Heun%27s_method" title="Heun's method">Heun's method</a></li> <li><a href="/wiki/Newmark-beta_method" title="Newmark-beta method">Newmark-beta method</a></li> <li><a href="/wiki/Leapfrog_integration" title="Leapfrog integration">Leapfrog integration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_methods_for_ordinary_differential_equations#Consistency_and_order" title="Numerical methods for ordinary differential equations">Higher-order methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_integrator" title="Exponential integrator">Exponential integrator</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/List_of_Runge%E2%80%93Kutta_methods" title="List of Runge–Kutta methods">List of Runge–Kutta methods</a></li> <li><a href="/wiki/Linear_multistep_method" title="Linear multistep method">Linear multistep method</a></li> <li><a href="/wiki/General_linear_methods" title="General linear methods">General linear methods</a></li> <li><a href="/wiki/Backward_differentiation_formula" title="Backward differentiation formula">Backward differentiation formula</a></li> <li><a href="/wiki/Leapfrog_integration#Yoshida_algorithms" title="Leapfrog integration">Yoshida</a></li> <li><a href="/wiki/Gauss%E2%80%93Legendre_method" title="Gauss–Legendre method">Gauss–Legendre method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Symplectic_integrator" title="Symplectic integrator">Symplectic integrator</a></li></ul> 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