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<span>Three-body problem</span> </div> </a> <ul id="toc-Three-body_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standardisation_of_astronomical_tables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standardisation_of_astronomical_tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Standardisation of astronomical tables</span> </div> </a> <ul id="toc-Standardisation_of_astronomical_tables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anomalous_precession_of_Mercury" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anomalous_precession_of_Mercury"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Anomalous precession of Mercury</span> </div> </a> <ul id="toc-Anomalous_precession_of_Mercury-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples_of_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples of problems</span> </div> </a> <ul id="toc-Examples_of_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Perturbation_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Perturbation_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Perturbation theory</span> </div> </a> <ul id="toc-Perturbation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference_frame" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Reference frame</span> </div> </a> <ul id="toc-Reference_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbital_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Orbital mechanics</span> </div> </a> <ul id="toc-Orbital_mechanics-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">6</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">7</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">8</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">9</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">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Celestial mechanics</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 64 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-64" 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">64 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hemelmeganika" title="Hemelmeganika – Afrikaans" lang="af" hreflang="af" data-title="Hemelmeganika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%B3%D9%85%D8%A7%D9%88%D9%8A%D8%A9" title="ميكانيكا سماوية – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا سماوية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mec%C3%A1nica_celeste" title="Mecánica celeste – Asturian" lang="ast" hreflang="ast" data-title="Mecánica celeste" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/G%C3%B6y_mexanikas%C4%B1" title="Göy mexanikası – Azerbaijani" lang="az" hreflang="az" data-title="Göy mexanikası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D1%8F%D0%B1%D0%B5%D1%81%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Нябесная механіка – Belarusian" lang="be" hreflang="be" data-title="Нябесная механіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9D%D1%8F%D0%B1%D0%B5%D1%81%D0%BD%D0%B0%D1%8F_%D0%BC%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Нябесная мэханіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Нябесная мэханіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Небесна механика – Bulgarian" lang="bg" hreflang="bg" data-title="Небесна механика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_celeste" title="Mecànica celeste – Catalan" lang="ca" hreflang="ca" data-title="Mecànica celeste" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D3%B3%D0%BF%D0%B5%D1%80%D0%B8_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Тӳпери механика – Chuvash" lang="cv" hreflang="cv" data-title="Тӳпери механика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nebesk%C3%A1_mechanika" title="Nebeská mechanika – Czech" lang="cs" hreflang="cs" data-title="Nebeská mechanika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Himmelmekanik" title="Himmelmekanik – Danish" lang="da" hreflang="da" data-title="Himmelmekanik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Himmelsmechanik" title="Himmelsmechanik – German" lang="de" hreflang="de" data-title="Himmelsmechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Taevamehaanika" title="Taevamehaanika – Estonian" lang="et" hreflang="et" data-title="Taevamehaanika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%85%CF%81%CE%AC%CE%BD%CE%B9%CE%B1_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Ουράνια μηχανική – Greek" lang="el" hreflang="el" data-title="Ουράνια μηχανική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_celeste" title="Mecánica celeste – Spanish" lang="es" hreflang="es" data-title="Mecánica celeste" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/%C4%88iela_mekaniko" title="Ĉiela mekaniko – Esperanto" lang="eo" hreflang="eo" data-title="Ĉiela mekaniko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zeruko_mekanika" title="Zeruko mekanika – Basque" lang="eu" hreflang="eu" data-title="Zeruko mekanika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D8%B3%D9%85%D8%A7%D9%88%DB%8C" title="مکانیک سماوی – Persian" lang="fa" hreflang="fa" data-title="مکانیک سماوی" 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/M%C3%A9canique_c%C3%A9leste" title="Mécanique céleste – French" lang="fr" hreflang="fr" data-title="Mécanique céleste" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_celeste" title="Mecánica celeste – Galician" lang="gl" hreflang="gl" data-title="Mecánica celeste" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%9C%EC%B2%B4%EC%97%AD%ED%95%99" title="천체역학 – Korean" lang="ko" hreflang="ko" data-title="천체역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D5%B6%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Երկնային մեխանիկա – Armenian" lang="hy" hreflang="hy" data-title="Երկնային մեխանիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%96%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A5%80%E0%A4%AF_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="खगोलीय यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="खगोलीय यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Nebeska_mehanika" title="Nebeska mehanika – Croatian" lang="hr" hreflang="hr" data-title="Nebeska mehanika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cielala_mekaniko" title="Cielala mekaniko – Ido" lang="io" hreflang="io" data-title="Cielala mekaniko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_benda_langit" title="Mekanika benda langit – Indonesian" lang="id" hreflang="id" data-title="Mekanika benda langit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Mechanica_celestial" title="Mechanica celestial – Interlingua" lang="ia" hreflang="ia" data-title="Mechanica celestial" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_celeste" title="Meccanica celeste – Italian" lang="it" hreflang="it" data-title="Meccanica celeste" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AA%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90" title="ცის მექანიკა – Georgian" lang="ka" hreflang="ka" data-title="ცის მექანიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%81%D0%BF%D0%B0%D0%BD_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D1%81%D1%8B" title="Аспан механикасы – Kazakh" lang="kk" hreflang="kk" data-title="Аспан механикасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umakanika_wa_anga" title="Umakanika wa anga – Swahili" lang="sw" hreflang="sw" data-title="Umakanika wa anga" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Himmelsmechanik" title="Himmelsmechanik – Luxembourgish" lang="lb" hreflang="lb" data-title="Himmelsmechanik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dangaus_mechanika" title="Dangaus mechanika – Lithuanian" lang="lt" hreflang="lt" data-title="Dangaus mechanika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%89gi_mechanika" title="Égi mechanika – Hungarian" lang="hu" hreflang="hu" data-title="Égi mechanika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Небесна механика – Macedonian" lang="mk" hreflang="mk" data-title="Небесна механика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%B3%D9%85%D8%A7%D9%88%D9%8A%D9%87" title="ميكانيكا سماويه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="ميكانيكا سماويه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Mekanika_bando_langik" title="Mekanika bando langik – Minangkabau" lang="min" hreflang="min" data-title="Mekanika bando langik" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Ti%C4%95ng-t%C4%81%CC%A4_l%C4%ADk-h%C5%8Fk" title="Tiĕng-tā̤ lĭk-hŏk – Mindong" lang="cdo" hreflang="cdo" data-title="Tiĕng-tā̤ lĭk-hŏk" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hemelmechanica" title="Hemelmechanica – Dutch" lang="nl" hreflang="nl" data-title="Hemelmechanica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%A9%E4%BD%93%E5%8A%9B%E5%AD%A6" title="天体力学 – Japanese" lang="ja" hreflang="ja" data-title="天体力学" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Himmelmekanikk" title="Himmelmekanikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Himmelmekanikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Celestmekanikk" title="Celestmekanikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Celestmekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Samoviy_mexanika" title="Samoviy mexanika – Uzbek" lang="uz" hreflang="uz" data-title="Samoviy mexanika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_nieba" title="Mechanika nieba – Polish" lang="pl" hreflang="pl" data-title="Mechanika nieba" 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/Mec%C3%A2nica_celeste" title="Mecânica celeste – Portuguese" lang="pt" hreflang="pt" data-title="Mecânica celeste" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mecanic%C4%83_cereasc%C4%83" title="Mecanică cerească – Romanian" lang="ro" hreflang="ro" data-title="Mecanică cerească" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Небесная механика – Russian" lang="ru" hreflang="ru" data-title="Небесная механика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Celestial_mechanics" title="Celestial mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Celestial mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Nebesk%C3%A1_mechanika" title="Nebeská mechanika – Slovak" lang="sk" hreflang="sk" data-title="Nebeská mechanika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Nebesna_mehanika" title="Nebesna mehanika – Slovenian" lang="sl" hreflang="sl" data-title="Nebesna mehanika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C_%D8%A6%D8%A7%D8%B3%D9%85%D8%A7%D9%86%DB%8C" title="میکانیکی ئاسمانی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="میکانیکی ئاسمانی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BA%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Небеска механика – Serbian" lang="sr" hreflang="sr" data-title="Небеска механика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Nebeska_mehanika" title="Nebeska mehanika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Nebeska mehanika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Taivaanmekaniikka" title="Taivaanmekaniikka – Finnish" lang="fi" hreflang="fi" data-title="Taivaanmekaniikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Celest_mekanik" title="Celest mekanik – Swedish" lang="sv" hreflang="sv" data-title="Celest mekanik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D1%81%D1%8B" title="Күк механикасы – Tatar" lang="tt" hreflang="tt" data-title="Күк механикасы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%97%E0%B9%89%E0%B8%AD%E0%B8%87%E0%B8%9F%E0%B9%89%E0%B8%B2" title="กลศาสตร์ท้องฟ้า – Thai" lang="th" hreflang="th" data-title="กลศาสตร์ท้องฟ้า" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/G%C3%B6k_mekani%C4%9Fi" title="Gök mekaniği – Turkish" lang="tr" hreflang="tr" data-title="Gök mekaniği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B1%D0%B5%D1%81%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Небесна механіка – Ukrainian" lang="uk" hreflang="uk" data-title="Небесна механіка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_thi%C3%AAn_th%E1%BB%83" title="Cơ học thiên thể – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ học thiên thể" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%A4%A9%E9%AB%94%E5%8A%9B%E5%AD%B8" title="天體力學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="天體力學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%A9%E4%BD%93%E5%8A%9B%E5%AD%A6" title="天体力学 – Wu" lang="wuu" hreflang="wuu" data-title="天体力学" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%A9%E9%AB%94%E5%8A%9B%E5%AD%B8" title="天體力學 – Cantonese" lang="yue" 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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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link 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" style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><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 {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a class="mw-selflink selflink">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a 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Neumann</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link 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 class="mw-selflink selflink"><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 href="/wiki/N-body_problem" title="N-body problem">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"><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: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><b>Celestial mechanics</b> is the <a href="/wiki/History_of_astronomy" title="History of astronomy">branch of</a> <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> that deals with the <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motions</a> of <a href="/wiki/Celestial_object" class="mw-redirect" title="Celestial object">objects in outer space</a>. Historically, celestial mechanics applies principles of <a href="/wiki/Physics" title="Physics">physics</a> (<a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>) to astronomical objects, such as <a href="/wiki/Star" title="Star">stars</a> and <a href="/wiki/Planet" title="Planet">planets</a>, to produce <a href="/wiki/Ephemeris" title="Ephemeris">ephemeris</a> data. </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=Celestial_mechanics&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">For early theories of the causes of planetary motion, see <a href="/wiki/Dynamics_of_the_celestial_spheres" title="Dynamics of the celestial spheres">Dynamics of the celestial spheres</a>.</div> <p>Modern analytic celestial mechanics started with <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>'s <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica"><i>Principia</i> (1687)</a>. The name <b>celestial mechanics</b> is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>, and over a century after Newton, <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> introduced the term <i>celestial mechanics</i>. Prior to <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a>, there was little connection between exact, quantitative prediction of planetary positions, using <a href="/wiki/Greek_astronomy#Eudoxan_astronomy" class="mw-redirect" title="Greek astronomy">geometrical</a> or <a href="/wiki/Babylonian_astronomy#Neo-Babylonian_astronomy" title="Babylonian astronomy">numerical</a> techniques, and contemporary discussions of the physical causes of the planets' motion. </p> <div class="mw-heading mw-heading3"><h3 id="Laws_of_planetary_motion">Laws of planetary motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=2" title="Edit section: Laws of planetary motion"><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">For detailed treatments of how his laws of planetary motion can be used, see <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a> and <a href="/wiki/Kepler_problem" title="Kepler problem">Kepler problem</a>.</div> <p><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> as the first to closely integrate the predictive geometrical astronomy, which had been dominant from <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a> in the 2nd century to <a href="/wiki/Copernicus" class="mw-redirect" title="Copernicus">Copernicus</a>, with physical concepts to produce a <a href="/wiki/Astronomia_nova" title="Astronomia nova"><i>New Astronomy, Based upon Causes, or Celestial Physics</i></a> in 1609. His work led to the <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">laws of planetary orbits</a>, which he developed using his physical principles and the <a href="/wiki/Planet" title="Planet">planetary</a> observations made by <a href="/wiki/Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a>. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Newton developed his <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">law of gravitation</a> in 1686. </p> <div class="mw-heading mw-heading3"><h3 id="Newtonian_mechanics_and_universal_gravitation">Newtonian mechanics and universal gravitation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=3" title="Edit section: Newtonian mechanics and universal gravitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> is credited with introducing the idea that the motion of objects in the heavens, such as <a href="/wiki/Planet" title="Planet">planets</a>, the <a href="/wiki/Sun" title="Sun">Sun</a>, and the <a href="/wiki/Moon" title="Moon">Moon</a>, and the motion of objects on the ground, like <a href="/wiki/Cannon" title="Cannon">cannon</a> balls and falling apples, could be described by the same set of <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical laws</a>. In this sense he unified <i>celestial</i> and <i>terrestrial</i> dynamics. Using <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">his law of gravity</a>, Newton confirmed <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a> for elliptical orbits by deriving them from the gravitational <a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a>, which Newton included in his epochal <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a></i> in 1687. </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=Celestial_mechanics&action=edit&section=4" 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>After Newton, <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> attempted to solve the three-body problem in 1772, analyzed the stability of planetary orbits, and discovered the existence of the <a href="/wiki/Lagrange_point" title="Lagrange point">Lagrange points</a>. Lagrange also reformulated the principles of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, emphasizing energy more than force, and developing a <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">method</a> to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and <a href="/wiki/Comet" title="Comet">comets</a> and such (parabolic and hyperbolic orbits are <a href="/wiki/Conic_section" title="Conic section">conic section</a> extensions of Kepler's <a href="/wiki/Elliptical_orbit" class="mw-redirect" title="Elliptical orbit">elliptical orbits</a>). More recently, it has also become useful to calculate <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a> <a href="/wiki/Trajectory" title="Trajectory">trajectories</a>. </p><p><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of <a href="/wiki/Algebra" title="Algebra">algebraic</a> and <a href="/wiki/Transcendental_functions" class="mw-redirect" title="Transcendental functions">transcendental functions</a> through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>These monographs include an idea of Poincaré, which later became the basis for mathematical "<a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>" (see, in particular, the <a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a>) and the general theory of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. He introduced the important concept of <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation points</a> and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).<sup id="cite_ref-Gold_Medal_to_Poincaré_2-0" class="reference"><a href="#cite_note-Gold_Medal_to_Poincaré-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Standardisation_of_astronomical_tables">Standardisation of astronomical tables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=5" title="Edit section: Standardisation of astronomical tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Simon_Newcomb" title="Simon Newcomb">Simon Newcomb</a> was a Canadian-American astronomer who revised <a href="/wiki/Peter_Andreas_Hansen" title="Peter Andreas Hansen">Peter Andreas Hansen</a>'s table of lunar positions. In 1877, assisted by <a href="/wiki/George_William_Hill" title="George William Hill">George William Hill</a>, he recalculated all the major astronomical constants. After 1884 he conceived, with A.M.W. Downing, a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in <a href="/wiki/Paris" title="Paris">Paris</a>, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard. </p> <div class="mw-heading mw-heading3"><h3 id="Anomalous_precession_of_Mercury">Anomalous precession of Mercury</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=6" title="Edit section: Anomalous precession of Mercury"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> explained the anomalous <a href="/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury" title="Tests of general relativity">precession of Mercury's perihelion</a> in his 1916 paper <i>The Foundation of the General Theory of Relativity</i>. <a href="/wiki/General_relativity" title="General relativity">General relativity</a> led astronomers to recognize that <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> did not provide the highest accuracy. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_problems">Examples of problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=7" title="Edit section: Examples of problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Celestial motion, without additional forces such as <a href="/wiki/Drag_force" class="mw-redirect" title="Drag force">drag forces</a> or the <a href="/wiki/Thrust" title="Thrust">thrust</a> of a <a href="/wiki/Rocket" title="Rocket">rocket</a>, is governed by the reciprocal gravitational acceleration between masses. A generalization is the <a href="/wiki/N-body_problem" title="N-body problem"><i>n</i>-body problem</a>,<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> where a number <i>n</i> of masses are mutually interacting via the gravitational force. Although analytically not <a href="/wiki/Integrable" class="mw-redirect" title="Integrable">integrable</a> in the general case,<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> the integration can be well approximated numerically. </p> <dl><dd>Examples: <ul><li>4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the <a href="/wiki/Patched_conic_approximation" title="Patched conic approximation">patched conic approximation</a>)</li> <li>3-body problem: <ul><li><a href="/wiki/Quasi-satellite" title="Quasi-satellite">Quasi-satellite</a></li> <li>Spaceflight to, and stay at a <a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian point</a></li></ul></li></ul></dd></dl> <p>In the <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 n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span> case (<a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a>) the configuration is much simpler than for <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 n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}"></span>. In this case, the system is fully integrable and exact solutions can be found.<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> <dl><dd>Examples: <ul><li>A <a href="/wiki/Binary_star" title="Binary star">binary star</a>, e.g., <a href="/wiki/Alpha_Centauri" title="Alpha Centauri">Alpha Centauri</a> (approx. the same mass)</li> <li>A <a href="/wiki/Binary_asteroid" title="Binary asteroid">binary asteroid</a>, e.g., <a href="/wiki/90_Antiope" title="90 Antiope">90 Antiope</a> (approx. the same mass)</li></ul></dd></dl> <p>A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the <a href="/wiki/Orbiting_body" title="Orbiting body">orbiting body</a>, is much smaller than the other, the <a href="/wiki/Central_body" class="mw-redirect" title="Central body">central body</a>. This is also often approximately valid. </p> <dl><dd>Examples: <ul><li>The <a href="/wiki/Solar_System" title="Solar System">Solar System</a> orbiting the center of the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a></li> <li>A planet orbiting the Sun</li> <li>A moon orbiting a planet</li> <li>A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)</li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Perturbation_theory">Perturbation theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=8" title="Edit section: Perturbation theory"><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/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></div> <p><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, which <a href="/wiki/Methods_of_computing_square_roots#Heron's_method" title="Methods of computing square roots">are ancient</a>.) The earliest use of modern <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a> was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>'s solution for the orbit of the <a href="/wiki/Moon" title="Moon">Moon</a>, which moves noticeably differently from a simple <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Keplerian ellipse</a> because of the competing gravitation of the <a href="/wiki/Earth" title="Earth">Earth</a> and the <a href="/wiki/Sun" title="Sun">Sun</a>. </p><p><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation methods</a> start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Keplerian ellipse</a>, which is correct when there are only two gravitating bodies (say, the <a href="/wiki/Earth" title="Earth">Earth</a> and the <a href="/wiki/Moon" title="Moon">Moon</a>), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use. </p><p>The solved, but simplified problem is then <i>"perturbed"</i> to make its <a href="/wiki/Differential_equation" title="Differential equation">time-rate-of-change equations for the object's position</a> closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the <a href="/wiki/Sun" title="Sun">Sun</a>). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem. </p><p>There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy. </p><p>The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> is reported to have said, regarding the problem of the <a href="/wiki/Moon" title="Moon">Moon</a>'s orbit <i>"It causeth my head to ache."</i><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> </p><p>This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method <a href="/wiki/Methods_of_computing_square_roots#Heron's_method" title="Methods of computing square roots">used anciently with numbers</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Reference_frame">Reference frame</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=9" title="Edit section: Reference frame"><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-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Celestial_mechanics&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">September 2023</span>)</i></span></div></td></tr></tbody></table> <p>Problems in celestial mechanics are often posed in simplifying reference frames, such as the <b>synodic reference frame</b> applied to the <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>, where the origin coincides with the <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> of the two larger celestial bodies. Other reference frames for n-body simulations include those that place the origin to follow the center of mass of a body, such as the heliocentric and the geocentric reference frames.<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> The choice of reference frame gives rise to many phenomena, including the <a href="/wiki/Retrograde_and_prograde_motion" title="Retrograde and prograde motion">retrograde motion</a> of <a href="/wiki/Inferior_and_superior_planets" title="Inferior and superior planets">superior planets</a> while on a geocentric reference frame. </p> <div class="mw-heading mw-heading2"><h2 id="Orbital_mechanics">Orbital mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=10" title="Edit section: Orbital mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital mechanics</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Orbital_mechanics&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbital_motion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Orbital_motion.gif/261px-Orbital_motion.gif" decoding="async" width="261" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4e/Orbital_motion.gif 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>A satellite orbiting Earth has a <a href="/wiki/Speed#Tangential_speed" title="Speed">tangential velocity</a> and an inward <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>.</figcaption></figure> <p><a href="/wiki/Orbital_mechanics" title="Orbital mechanics">Orbital mechanics</a> or astrodynamics is the application of <a href="/wiki/Ballistics" title="Ballistics">ballistics</a> and celestial mechanics to the practical problems concerning the motion of <a href="/wiki/Rocket" title="Rocket">rockets</a>, <a href="/wiki/Satellite" title="Satellite">satellites</a>, and other <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a>. The motion of these objects is usually calculated from <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> and the <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">law of universal gravitation</a>. Orbital mechanics is a core discipline within <a href="/wiki/Space_exploration" title="Space exploration">space-mission</a> design and control. </p><p>Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of <a href="/wiki/Gravity" title="Gravity">gravity</a>, including both spacecraft and natural <a href="/wiki/Astronomical_object" title="Astronomical object">astronomical bodies</a> such as <a href="/wiki/Star_system" title="Star system">star systems</a>, <a href="/wiki/Planet" title="Planet">planets</a>, <a href="/wiki/Natural_satellite" title="Natural satellite">moons</a>, and <a href="/wiki/Comet" title="Comet">comets</a>. Orbital mechanics focuses on spacecraft <a href="/wiki/Trajectory" title="Trajectory">trajectories</a>, including <a href="/wiki/Orbital_maneuver" title="Orbital maneuver">orbital maneuvers</a>, <a href="/wiki/Orbital_plane_(astronomy)" class="mw-redirect" title="Orbital plane (astronomy)">orbital plane</a> changes, and interplanetary transfers, and is used by mission planners to predict the results of <a href="/wiki/Spacecraft_propulsion" title="Spacecraft propulsion">propulsive maneuvers</a>. </p> <a href="/wiki/General_relativity" title="General relativity">General relativity</a> is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).</div></div> <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=Celestial_mechanics&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Astrometry" title="Astrometry">Astrometry</a> is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.</li> <li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Celestial_navigation" title="Celestial navigation">Celestial navigation</a> is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.</li> <li><a href="/wiki/Jet_Propulsion_Laboratory_Developmental_Ephemeris" class="mw-redirect" title="Jet Propulsion Laboratory Developmental Ephemeris">Developmental Ephemeris</a> or the <a href="/wiki/Jet_Propulsion_Laboratory_Developmental_Ephemeris" class="mw-redirect" title="Jet Propulsion Laboratory Developmental Ephemeris">Jet Propulsion Laboratory Developmental Ephemeris</a> (JPL DE) is a widely used model of the solar system, which combines celestial mechanics with <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and astronomical and spacecraft data.</li> <li><a href="/wiki/Dynamics_of_the_celestial_spheres" title="Dynamics of the celestial spheres">Dynamics of the celestial spheres</a> concerns pre-Newtonian explanations of the causes of the motions of the stars and planets.</li> <li><a href="/wiki/Dynamical_time_scale" title="Dynamical time scale">Dynamical time scale</a></li> <li><a href="/wiki/Ephemeris" title="Ephemeris">Ephemeris</a> is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.</li> <li><a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">Gravitation</a></li> <li><a href="/wiki/Lunar_theory" title="Lunar theory">Lunar theory</a> attempts to account for the motions of the Moon.</li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a <a href="/wiki/Planet" title="Planet">planet</a> in the sky) which are too difficult to solve down to a general, exact formula.</li> <li>Creating a <a href="/wiki/Numerical_model_of_solar_system" class="mw-redirect" title="Numerical model of solar system">numerical model of the solar system</a> was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.</li> <li>An <i><a href="/wiki/Orbit" title="Orbit">orbit</a></i> is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.</li> <li><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a> are the parameters needed to specify a Newtonian two-body orbit uniquely.</li> <li><a href="/wiki/Osculating_orbit" title="Osculating orbit">Osculating orbit</a> is the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.</li> <li><a href="/wiki/Retrograde_motion" class="mw-redirect" title="Retrograde motion">Retrograde motion</a> is orbital motion in a system, such as a planet and its satellites, that is contrary to the direction of rotation of the central body, or more generally contrary in direction to the net angular momentum of the entire system.</li> <li><a href="/wiki/Apparent_retrograde_motion" title="Apparent retrograde motion">Apparent retrograde motion</a> is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).</li> <li><a href="/wiki/Satellite" title="Satellite">Satellite</a> is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons). The common noun ‘moon’ (not capitalized) is used to mean any <a href="/wiki/Natural_satellite" title="Natural satellite">natural satellite</a> of the other planets.</li> <li><a href="/wiki/Tidal_force" title="Tidal force">Tidal force</a> is the combination of out-of-balance forces and accelerations of (mostly) solid bodies that raises tides in bodies of liquid (oceans), atmospheres, and strains planets' and satellites' crusts.</li> <li>Two solutions, called <a href="/wiki/VSOP_(planets)" class="mw-redirect" title="VSOP (planets)">VSOP82 and VSOP87</a> are versions one mathematical theory for the orbits and positions of the major planets, which seeks to provide accurate positions over an extended period of time.</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=Celestial_mechanics&action=edit&section=12" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">J. Stillwell, Mathematics and its history, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254">page 254</a></span> </li> <li id="cite_note-Gold_Medal_to_Poincaré-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gold_Medal_to_Poincaré_2-0">^</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="CITEREFDarwin1900" class="citation journal cs1">Darwin, G.H. (1900). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2F60.5.406">"Address Delivered by the President, Professor G. H. Darwin, on presenting the Gold Medal of the Society to M. H. Poincaré"</a>. <i>Monthly Notices of the Royal Astronomical Society</i>. <b>60</b> (5): <span class="nowrap">406–</span>416. <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/10.1093%2Fmnras%2F60.5.406">10.1093/mnras/60.5.406</a></span>. <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/0035-8711">0035-8711</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=Address+Delivered+by+the+President%2C+Professor+G.+H.+Darwin%2C+on+presenting+the+Gold+Medal+of+the+Society+to+M.+H.+Poincar%C3%A9&rft.volume=60&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E406-%3C%2Fspan%3E416&rft.date=1900&rft_id=info%3Adoi%2F10.1093%2Fmnras%2F60.5.406&rft.issn=0035-8711&rft.aulast=Darwin&rft.aufirst=G.H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fmnras%252F60.5.406&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrentiHut2008" class="citation journal cs1">Trenti, Michele; Hut, Piet (2008-05-20). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.3930">"N-body simulations (gravitational)"</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>. <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/1941-6016">1941-6016</a>.</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=N-body+simulations+%28gravitational%29&rft.volume=3&rft.issue=5&rft.pages=3930&rft.date=2008-05-20&rft.issn=1941-6016&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%3ACelestial+mechanics" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCombot2015" class="citation arxiv cs1">Combot, Thierry (2015-09-01). "Integrability and non integrability of some n body problems". <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/1509.08233">1509.08233</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.DS">math.DS</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Integrability+and+non+integrability+of+some+n+body+problems&rft.date=2015-09-01&rft_id=info%3Aarxiv%2F1509.08233&rft.aulast=Combot&rft.aufirst=Thierry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://scienceworld.wolfram.com/physics/Two-BodyProblem.html">"Two-Body Problem -- from Eric Weisstein's World of Physics"</a>. <i>scienceworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=scienceworld.wolfram.com&rft.atitle=Two-Body+Problem+--+from+Eric+Weisstein%27s+World+of+Physics&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FTwo-BodyProblem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCropper2004" class="citation cs2">Cropper, William H. (2004), <i>Great Physicists: The life and times of leading physicists from Galileo to Hawking</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, p. 34, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-517324-6" title="Special:BookSources/978-0-19-517324-6"><bdi>978-0-19-517324-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Great+Physicists%3A+The+life+and+times+of+leading+physicists+from+Galileo+to+Hawking&rft.pages=34&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=978-0-19-517324-6&rft.aulast=Cropper&rft.aufirst=William+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuerraCarvalho2016" class="citation journal cs1">Guerra, André G C; Carvalho, Paulo Simeão (1 August 2016). "Orbital motions of astronomical bodies and their centre of mass from different reference frames: a conceptual step between the geocentric and heliocentric models". <i>Physics Education</i>. <b>51</b> (5). <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/1605.01339">1605.01339</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/2016PhyEd..51e5012G">2016PhyEd..51e5012G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9120%2F51%2F5%2F055012">10.1088/0031-9120/51/5/055012</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Education&rft.atitle=Orbital+motions+of+astronomical+bodies+and+their+centre+of+mass+from+different+reference+frames%3A+a+conceptual+step+between+the+geocentric+and+heliocentric+models&rft.volume=51&rft.issue=5&rft.date=2016-08-01&rft_id=info%3Aarxiv%2F1605.01339&rft_id=info%3Adoi%2F10.1088%2F0031-9120%2F51%2F5%2F055012&rft_id=info%3Abibcode%2F2016PhyEd..51e5012G&rft.aulast=Guerra&rft.aufirst=Andr%C3%A9+G+C&rft.au=Carvalho%2C+Paulo+Sime%C3%A3o&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Celestial_mechanics&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Forest R. Moulton, <i>Introduction to Celestial Mechanics</i>, 1984, Dover, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-64687-4" title="Special:BookSources/0-486-64687-4">0-486-64687-4</a></li> <li>John E. Prussing, Bruce A. Conway, <i>Orbital Mechanics</i>, 1993, Oxford Univ. Press</li> <li>William M. Smart, <i>Celestial Mechanics</i>, 1961, John Wiley.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoggett1997" class="citation cs2">Doggett, LeRoy E. (1997), "Celestial Mechanics", in Lankford, John (ed.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fIzMMe3VczkC"><i>History of Astronomy: An Encyclopedia</i></a>, New York: Taylor & Francis, pp. <span class="nowrap">131–</span>140, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780815303220" title="Special:BookSources/9780815303220"><bdi>9780815303220</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Celestial+Mechanics&rft.btitle=History+of+Astronomy%3A+An+Encyclopedia&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E131-%3C%2Fspan%3E140&rft.pub=Taylor+%26+Francis&rft.date=1997&rft.isbn=9780815303220&rft.aulast=Doggett&rft.aufirst=LeRoy+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfIzMMe3VczkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></li> <li>J.M.A. Danby, <i>Fundamentals of Celestial Mechanics</i>, 1992, Willmann-Bell</li> <li>Alessandra Celletti, Ettore Perozzi, <i>Celestial Mechanics: The Waltz of the Planets</i>, 2007, Springer-Praxis, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-30777-X" title="Special:BookSources/0-387-30777-X">0-387-30777-X</a>.</li> <li>Michael Efroimsky. 2005. <i>Gauge Freedom in Orbital Mechanics.</i> <a rel="nofollow" class="external text" href="https://archive.today/20130105074903/http://www3.interscience.wiley.com/journal/118692589/abstract?CRETRY=1&SRETRY=0">Annals of the New York Academy of Sciences, Vol. 1065, pp. 346-374</a></li> <li>Alessandra Celletti, <i>Stability and Chaos in Celestial Mechanics.</i> Springer-Praxis 2010, XVI, 264 p., Hardcover <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-85145-5" title="Special:BookSources/978-3-540-85145-5">978-3-540-85145-5</a></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=Celestial_mechanics&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Encyclopedia:Celestial_mechanics">Encyclopedia:Celestial mechanics</a> <a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a> Expert articles</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1967" class="citation book cs1">Poincaré, H. (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/newmethodsofcele0000poin"><i>New Methods of Celestial Mechanics</i></a></span> (3 vol. English translated ed.). American Institute of Physics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56396-117-5" title="Special:BookSources/978-1-56396-117-5"><bdi>978-1-56396-117-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+Methods+of+Celestial+Mechanics&rft.edition=3+vol.+English+translated&rft.pub=American+Institute+of+Physics&rft.date=1967&rft.isbn=978-1-56396-117-5&rft.aulast=Poincar%C3%A9&rft.aufirst=H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnewmethodsofcele0000poin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></li></ul> <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=Celestial_mechanics&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Celestial_mechanics" class="extiw" title="commons:Category:Celestial mechanics">Celestial mechanics</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCalvert2003" class="citation cs2">Calvert, James B. (2003-03-28), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060907120741/http://www.du.edu/~jcalvert/phys/orbits.htm"><i>Celestial Mechanics</i></a>, University of Denver, archived from <a rel="nofollow" class="external text" href="http://www.du.edu/~jcalvert/phys/orbits.htm">the original</a> on 2006-09-07<span class="reference-accessdate">, retrieved <span class="nowrap">2006-08-21</span></span></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=University+of+Denver&rft.date=2003-03-28&rft.aulast=Calvert&rft.aufirst=James+B.&rft_id=http%3A%2F%2Fwww.du.edu%2F~jcalvert%2Fphys%2Forbits.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACelestial+mechanics" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.phy6.org/stargaze/Sastron.htm">Astronomy of the Earth's Motion in Space</a>, high-school level educational web site by David P. Stern</li> <li><a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/336k/Newton.pdf">Newtonian Dynamics</a> Undergraduate level course by Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the 3-body problem with the Sun, Moon, and the Earth).</li></ul> <p><b>Research</b> </p> <ul><li><a rel="nofollow" class="external text" href="http://www.math.washington.edu/~hampton/research.html">Marshall Hampton's research page: Central configurations in the n-body problem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20021001170939/http://www.math.washington.edu/~hampton/research.html">Archived</a> 2002-10-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <p><b>Artwork</b> </p> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190428072152/http://www.cmlab.com/">Celestial Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne</a></li></ul> <p><b>Course notes</b> </p> <ul><li><a rel="nofollow" class="external text" href="http://orca.phys.uvic.ca/~tatum/celmechs.html">Professor Tatum's course notes at the University of Victoria</a></li></ul> <p><b>Associations</b> </p> <ul><li><a rel="nofollow" class="external text" href="http://www.mat.uniroma2.it/simca/english.html">Italian Celestial Mechanics and Astrodynamics Association</a></li></ul> <p><b>Simulations</b> </p> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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