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spatial extent</span> </div> </a> <ul id="toc-Bodies_with_spatial_extent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vector_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vector form</span> </div> </a> <ul id="toc-Vector_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gravity_field" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gravity_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Gravity field</span> </div> </a> <ul id="toc-Gravity_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Limitations</span> </div> </a> <button aria-controls="toc-Limitations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Limitations subsection</span> </button> <ul id="toc-Limitations-sublist" class="vector-toc-list"> <li id="toc-Observations_conflicting_with_Newton's_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Observations_conflicting_with_Newton's_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Observations conflicting with Newton's formula</span> </div> </a> <ul id="toc-Observations_conflicting_with_Newton's_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einstein's_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein's_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Einstein's solution</span> </div> </a> <ul id="toc-Einstein's_solution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Extensions</span> </div> </a> <ul id="toc-Extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Solutions</span> </div> </a> <ul id="toc-Solutions-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">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <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" > <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">Newton's law of universal gravitation</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 90 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-90" 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">90 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/Newton_se_Swaartekragwet" title="Newton se Swaartekragwet – Afrikaans" lang="af" hreflang="af" data-title="Newton se Swaartekragwet" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%92%E1%8B%8D%E1%89%B0%E1%8A%95_%E1%8B%A8%E1%8C%8D%E1%88%B5%E1%89%A0%E1%89%B5_%E1%89%80%E1%88%98%E1%88%AD" title="የኒውተን የግስበት ቀመር – Amharic" lang="am" hreflang="am" data-title="የኒውተን የግስበት ቀመር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%A7%D9%84%D8%AC%D8%B0%D8%A8_%D8%A7%D9%84%D8%B9%D8%A7%D9%85_%D9%84%D9%86%D9%8A%D9%88%D8%AA%D9%86" title="قانون الجذب العام لنيوتن – Arabic" lang="ar" hreflang="ar" data-title="قانون الجذب العام لنيوتن" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%89%E0%A6%9F%E0%A6%A8%E0%A7%B0_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%AC%E0%A6%9C%E0%A6%A8%E0%A7%80%E0%A6%A8_%E0%A6%AE%E0%A6%B9%E0%A6%BE%E0%A6%95%E0%A7%B0%E0%A7%8D%E0%A6%B7%E0%A6%A3_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="নিউটনৰ বিশ্বজনীন মহাকৰ্ষণ সূত্ৰ – Assamese" lang="as" hreflang="as" data-title="নিউটনৰ বিশ্বজনীন মহাকৰ্ষণ সূত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llei_de_gravitaci%C3%B3n_universal" title="Llei de gravitación universal – Asturian" lang="ast" hreflang="ast" data-title="Llei de gravitación universal" 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/%C3%9Cmumd%C3%BCnya_cazib%C9%99_qanunu" title="Ümumdünya cazibə qanunu – Azerbaijani" lang="az" hreflang="az" data-title="Ümumdünya cazibə qanunu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D8%AA%D9%88%D9%86%D9%88%D9%86_%D8%AF%D9%88%D9%86%DB%8C%D8%A7%D9%84%DB%8C%D9%82_%D8%AC%D8%A7%D8%B0%D8%A8%D9%87_%D9%82%D8%A7%D9%86%D9%88%D9%86%D9%88" title="نیوتونون دونیالیق جاذبه قانونو – South Azerbaijani" lang="azb" hreflang="azb" data-title="نیوتونون دونیالیق جاذبه قانونو" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%89%E0%A6%9F%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%AE%E0%A6%B9%E0%A6%BE%E0%A6%95%E0%A6%B0%E0%A7%8D%E0%A6%B7_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="নিউটনের মহাকর্ষ সূত্র – Bangla" lang="bn" hreflang="bn" data-title="নিউটনের মহাকর্ষ সূত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B4%D1%8B%D2%A3_%D0%BA%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D0%BA_%D1%82%D0%B0%D1%80%D1%82%D1%8B%D0%BB%D1%8B%D1%83_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Ньютондың классик тартылыу теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Ньютондың классик тартылыу теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D1%81%D1%83%D1%81%D0%B2%D0%B5%D1%82%D0%BD%D0%B0%D0%B3%D0%B0_%D0%BF%D1%80%D1%8B%D1%86%D1%8F%D0%B3%D0%BD%D0%B5%D0%BD%D0%BD%D1%8F" title="Закон сусветнага прыцягнення – Belarusian" lang="be" hreflang="be" data-title="Закон сусветнага прыцягнення" 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%9A%D0%BB%D1%8F%D1%81%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%BF%D1%80%D1%8B%D1%86%D1%8F%D0%B3%D0%BD%D0%B5%D0%BD%D1%8C%D0%BD%D1%8F_%D0%9D%D1%8C%D1%8E%D1%82%D0%B0%D0%BD%D0%B0" title="Клясычная тэорыя прыцягненьня Ньютана – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Клясычная тэорыя прыцягненьня Ньютана" 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%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B7%D0%B0_%D0%B2%D1%81%D0%B5%D0%BE%D0%B1%D1%89%D0%BE%D1%82%D0%BE_%D0%BF%D1%80%D0%B8%D0%B2%D0%BB%D0%B8%D1%87%D0%B0%D0%BD%D0%B5" title="Закон за всеобщото привличане – Bulgarian" lang="bg" hreflang="bg" data-title="Закон за всеобщото привличане" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije – Bosnian" lang="bs" hreflang="bs" data-title="Newtonov zakon gravitacije" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_la_gravitaci%C3%B3_universal" title="Llei de la gravitació universal – Catalan" lang="ca" hreflang="ca" data-title="Llei de la gravitació universal" 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%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%C4%83%D0%BD_%D1%82%D1%83%D1%80%D1%82%C4%83%D0%BC_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Ньютонăн туртăм теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Ньютонăн туртăм теорийĕ" 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/Newton%C5%AFv_gravita%C4%8Dn%C3%AD_z%C3%A1kon" title="Newtonův gravitační zákon – Czech" lang="cs" hreflang="cs" data-title="Newtonův gravitační zákon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deddf_disgyrchedd_cyffredinol_Newton" title="Deddf disgyrchedd cyffredinol Newton – Welsh" lang="cy" hreflang="cy" data-title="Deddf disgyrchedd cyffredinol Newton" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Newtonsk_gravitation" title="Newtonsk gravitation – Danish" lang="da" hreflang="da" data-title="Newtonsk gravitation" 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/Newtonsches_Gravitationsgesetz" title="Newtonsches Gravitationsgesetz – German" lang="de" hreflang="de" data-title="Newtonsches Gravitationsgesetz" 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/Gravitatsiooniseadus" title="Gravitatsiooniseadus – Estonian" lang="et" hreflang="et" data-title="Gravitatsiooniseadus" 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%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CE%B7%CF%82_%CF%80%CE%B1%CE%B3%CE%BA%CF%8C%CF%83%CE%BC%CE%B9%CE%B1%CF%82_%CE%AD%CE%BB%CE%BE%CE%B7%CF%82" title="Νόμος της παγκόσμιας έλξης – Greek" lang="el" hreflang="el" data-title="Νόμος της παγκόσμιας έλξης" 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/Ley_de_gravitaci%C3%B3n_universal" title="Ley de gravitación universal – Spanish" lang="es" hreflang="es" data-title="Ley de gravitación universal" 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/Ne%C5%ADtona_le%C4%9Do_pri_universala_gravito" title="Neŭtona leĝo pri universala gravito – Esperanto" lang="eo" hreflang="eo" data-title="Neŭtona leĝo pri universala gravito" 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/Grabitazio_unibertsalaren_legea" title="Grabitazio unibertsalaren legea – Basque" lang="eu" hreflang="eu" data-title="Grabitazio unibertsalaren legea" 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%82%D8%A7%D9%86%D9%88%D9%86_%D8%AC%D9%87%D8%A7%D9%86%DB%8C_%DA%AF%D8%B1%D8%A7%D9%86%D8%B4_%D9%86%DB%8C%D9%88%D8%AA%D9%86" title="قانون جهانی گرانش نیوتن – Persian" lang="fa" hreflang="fa" data-title="قانون جهانی گرانش نیوتن" 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/Loi_universelle_de_la_gravitation" title="Loi universelle de la gravitation – French" lang="fr" hreflang="fr" data-title="Loi universelle de la gravitation" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_na_himtharraingthe" title="Dlí na himtharraingthe – Irish" lang="ga" hreflang="ga" data-title="Dlí na himtharraingthe" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Lei_da_gravitaci%C3%B3n_universal" title="Lei da gravitación universal – Galician" lang="gl" hreflang="gl" data-title="Lei da gravitación universal" 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/%EB%A7%8C%EC%9C%A0%EC%9D%B8%EB%A0%A5%EC%9D%98_%EB%B2%95%EC%B9%99" title="만유인력의 법칙 – Korean" lang="ko" hreflang="ko" data-title="만유인력의 법칙" 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/%D5%86%D5%B5%D5%B8%D6%82%D5%BF%D5%B8%D5%B6%D5%AB_%D5%A4%D5%A1%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B1%D5%A3%D5%B8%D5%B2%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Նյուտոնի դասական ձգողության տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Նյուտոնի դասական ձգողության տեսություն" 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%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="न्यूटन का सार्वत्रिक गुरुत्वाकर्षण का सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="न्यूटन का सार्वत्रिक गुरुत्वाकर्षण का सिद्धान्त" 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/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije – Croatian" lang="hr" hreflang="hr" data-title="Newtonov zakon gravitacije" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_gravitasi_universal_Newton" title="Hukum gravitasi universal Newton – Indonesian" lang="id" hreflang="id" data-title="Hukum gravitasi universal Newton" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_di_gravitazione_universale" title="Legge di gravitazione universale – Italian" lang="it" hreflang="it" data-title="Legge di gravitazione universale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%94%D7%9B%D7%91%D7%99%D7%93%D7%94_%D7%94%D7%A2%D7%95%D7%9C%D7%9E%D7%99_%D7%A9%D7%9C_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F" title="חוק הכבידה העולמי של ניוטון – Hebrew" lang="he" hreflang="he" data-title="חוק הכבידה העולמי של ניוטון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A1%E1%83%9D%E1%83%A4%E1%83%9A%E1%83%98%E1%83%9D_%E1%83%9B%E1%83%98%E1%83%96%E1%83%98%E1%83%93%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="მსოფლიო მიზიდულობის კანონი – Georgian" lang="ka" hreflang="ka" data-title="მსოფლიო მიზიდულობის კანონი" 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%91%D2%AF%D0%BA%D1%96%D0%BB_%D3%99%D0%BB%D0%B5%D0%BC%D0%B4%D1%96%D0%BA_%D1%82%D0%B0%D1%80%D1%82%D1%8B%D0%BB%D1%8B%D1%81_%D0%B7%D0%B0%D2%A3%D1%8B" title="Бүкіл әлемдік тартылыс заңы – Kazakh" lang="kk" hreflang="kk" data-title="Бүкіл әлемдік тартылыс заңы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Lwa_iniv%C3%A8s%C3%A8l_gravitasyon" title="Lwa inivèsèl gravitasyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Lwa inivèsèl gravitasyon" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Lex_universalis_gravitationis_Newtoniana" title="Lex universalis gravitationis Newtoniana – Latin" lang="la" hreflang="la" data-title="Lex universalis gravitationis Newtoniana" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C5%85%C5%ABtona_vispasaules_gravit%C4%81cijas_likums" title="Ņūtona vispasaules gravitācijas likums – Latvian" lang="lv" hreflang="lv" data-title="Ņūtona vispasaules gravitācijas likums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Gravitatiounsgesetz" title="Gravitatiounsgesetz – Luxembourgish" lang="lb" hreflang="lb" data-title="Gravitatiounsgesetz" 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/Niutono_gravitacijos_d%C4%97snis" title="Niutono gravitacijos dėsnis – Lithuanian" lang="lt" hreflang="lt" data-title="Niutono gravitacijos dėsnis" 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/Newton-f%C3%A9le_gravit%C3%A1ci%C3%B3s_t%C3%B6rv%C3%A9ny" title="Newton-féle gravitációs törvény – Hungarian" lang="hu" hreflang="hu" data-title="Newton-féle gravitációs törvény" 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%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B7%D0%B0_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B0" title="Њутнов закон за гравитацијата – Macedonian" lang="mk" hreflang="mk" data-title="Њутнов закон за гравитацијата" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%90%E0%B4%B8%E0%B4%95%E0%B5%8D_%E0%B4%A8%E0%B5%8D%E0%B4%AF%E0%B5%82%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%86_%E0%B4%97%E0%B5%81%E0%B4%B0%E0%B5%81%E0%B4%A4%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%BC%E0%B4%B7%E0%B4%A3_%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="ഐസക് ന്യൂട്ടന്റെ ഗുരുത്വാകർഷണ നിയമം – Malayalam" lang="ml" hreflang="ml" data-title="ഐസക് ന്യൂട്ടന്റെ ഗുരുത്വാകർഷണ നിയമം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8%E0%A4%9A%E0%A4%BE_%E0%A4%B5%E0%A5%88%E0%A4%B6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BE%E0%A4%9A%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्यूटनचा वैश्विक गुरुत्वाकर्षणाचा नियम – Marathi" lang="mr" hreflang="mr" data-title="न्यूटनचा वैश्विक गुरुत्वाकर्षणाचा नियम" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A3%E1%83%9C%E1%83%98%E1%83%95%E1%83%94%E1%83%A0%E1%83%A1%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%A0%E1%83%90%E1%83%95%E1%83%98%E1%83%A2%E1%83%90%E1%83%AA%E1%83%98%E1%83%90%E1%83%A8_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="უნივერსალური გრავიტაციაშ კანონი – Mingrelian" lang="xmf" hreflang="xmf" data-title="უნივერსალური გრავიტაციაშ კანონი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum_kegravitian_semesta_Newton" title="Hukum kegravitian semesta Newton – Malay" lang="ms" hreflang="ms" data-title="Hukum kegravitian semesta Newton" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/U%C3%A2ng-i%C5%AB_%C4%ABng-l%C4%ADk_d%C3%AAng-l%C5%ADk" title="Uâng-iū īng-lĭk dêng-lŭk – Mindong" lang="cdo" hreflang="cdo" data-title="Uâng-iū īng-lĭk dêng-lŭk" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D2%AF%D1%85_%D0%B5%D1%80%D1%82%D3%A9%D0%BD%D1%86%D0%B8%D0%B9%D0%BD_%D1%82%D0%B0%D1%82%D0%B0%D0%BB%D1%86%D0%BB%D1%8B%D0%BD_%D1%85%D1%83%D1%83%D0%BB%D1%8C" title="Бүх ертөнцийн таталцлын хууль – Mongolian" lang="mn" hreflang="mn" data-title="Бүх ертөнцийн таталцлын хууль" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%94%E1%80%9A%E1%80%B0%E1%80%90%E1%80%94%E1%80%BA%E1%81%8F_%E1%80%85%E1%80%80%E1%80%BC%E1%80%9D%E1%80%A0%E1%80%AC%E1%80%86%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%9B%E1%80%AC_%E1%80%92%E1%80%BC%E1%80%95%E1%80%BA%E1%80%86%E1%80%BD%E1%80%B2%E1%80%A1%E1%80%AC%E1%80%B8%E1%80%94%E1%80%AD%E1%80%9A%E1%80%AC%E1%80%99" title="နယူတန်၏ စကြဝဠာဆိုင်ရာ ဒြပ်ဆွဲအားနိယာမ – Burmese" lang="my" hreflang="my" data-title="နယူတန်၏ စကြဝဠာဆိုင်ရာ ဒြပ်ဆွဲအားနိယာမ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gravitatiewet_van_Newton" title="Gravitatiewet van Newton – Dutch" lang="nl" hreflang="nl" data-title="Gravitatiewet van Newton" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9F%E0%A4%A8%E0%A4%95%E0%A5%8B_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%B8%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%A8%E0%A5%8D%E0%A4%A7%E0%A5%80_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B5%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AA%E0%A5%80_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्युटनको गुरुत्वाकर्षणसम्बन्धी विश्वव्यापी नियम – Nepali" lang="ne" hreflang="ne" data-title="न्युटनको गुरुत्वाकर्षणसम्बन्धी विश्वव्यापी नियम" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%87%E6%9C%89%E5%BC%95%E5%8A%9B" title="万有引力 – Japanese" lang="ja" hreflang="ja" data-title="万有引力" 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/Newtons_gravitasjonslov" title="Newtons gravitasjonslov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Newtons gravitasjonslov" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Lei_de_la_gravitacion_universala" title="Lei de la gravitacion universala – Occitan" lang="oc" hreflang="oc" data-title="Lei de la gravitacion universala" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%B0%E0%A9%82%E0%A8%A4%E0%A8%BE%E0%A8%95%E0%A8%B0%E0%A8%B8%E0%A8%BC%E0%A8%A3_%E0%A8%A6%E0%A8%BE_%E0%A8%B8%E0%A8%B0%E0%A8%B5-%E0%A8%B5%E0%A8%BF%E0%A8%85%E0%A8%BE%E0%A8%AA%E0%A9%80_%E0%A8%A8%E0%A8%BF%E0%A8%AF%E0%A8%AE" title="ਗੁਰੂਤਾਕਰਸ਼ਣ ਦਾ ਸਰਵ-ਵਿਅਾਪੀ ਨਿਯਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੁਰੂਤਾਕਰਸ਼ਣ ਦਾ ਸਰਵ-ਵਿਅਾਪੀ ਨਿਯਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D9%86_%D8%AF%D8%A7_%DA%A9%DA%BE%DA%86_%D8%AF%D8%A7_%D9%82%D9%86%D9%88%D9%86" title="نیوٹن دا کھچ دا قنون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نیوٹن دا کھچ دا قنون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D9%86%D9%8A%D9%88%D9%BC%D9%86_%D8%AF_%D9%86%DA%93%D9%8A%D9%88%D8%A7%D9%84%DB%90_%D8%AC%D8%A7%D8%B0%D8%A8%DB%90_%D9%82%D8%A7%D9%86%D9%88%D9%86" title="د نيوټن د نړيوالې جاذبې قانون – Pashto" lang="ps" hreflang="ps" data-title="د نيوټن د نړيوالې جاذبې قانون" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Lej_%C3%ABd_gravitassion_universal" title="Lej ëd gravitassion universal – Piedmontese" lang="pms" hreflang="pms" data-title="Lej ëd gravitassion universal" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawo_powszechnego_ci%C4%85%C5%BCenia" title="Prawo powszechnego ciążenia – Polish" lang="pl" hreflang="pl" data-title="Prawo powszechnego ciążenia" 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/Lei_da_gravita%C3%A7%C3%A3o_universal" title="Lei da gravitação universal – Portuguese" lang="pt" hreflang="pt" data-title="Lei da gravitação universal" 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/Legea_atrac%C8%9Biei_universale" title="Legea atracției universale – Romanian" lang="ro" hreflang="ro" data-title="Legea atracției universale" 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%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%82%D1%8F%D0%B3%D0%BE%D1%82%D0%B5%D0%BD%D0%B8%D1%8F_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0" title="Классическая теория тяготения Ньютона – Russian" lang="ru" hreflang="ru" data-title="Классическая теория тяготения Ньютона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Newton%27s_law_o_universal_gravitation" title="Newton's law o universal gravitation – Scots" lang="sco" hreflang="sco" data-title="Newton's law o universal gravitation" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ligji_gravitacional_universal_i_Njutonit" title="Ligji gravitacional universal i Njutonit – Albanian" lang="sq" hreflang="sq" data-title="Ligji gravitacional universal i Njutonit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%BB%E0%B7%8A%E0%B7%80%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B6%9C%E0%B7%94%E0%B6%BB%E0%B7%94%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%8F%E0%B6%9A%E0%B6%BB%E0%B7%8A%E0%B7%82%E0%B6%AB%E0%B6%BA_%E0%B6%B4%E0%B7%92%E0%B7%85%E0%B7%92%E0%B6%B6%E0%B6%B3_%E0%B6%B1%E0%B7%92%E0%B7%80%E0%B7%8A%E0%B6%A7%E0%B6%B1%E0%B7%8A%E0%B6%9C%E0%B7%9A_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%B8" title="සර්වත්‍ර ගුරුත්වාකර්ෂණය පිළිබඳ නිව්ටන්ගේ නියම – Sinhala" lang="si" hreflang="si" data-title="සර්වත්‍ර ගුරුත්වාකර්ෂණය පිළිබඳ නිව්ටන්ගේ නියම" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Newton's law of universal gravitation" 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/Gravita%C4%8Dn%C3%BD_z%C3%A1kon" title="Gravitačný zákon – Slovak" lang="sk" hreflang="sk" data-title="Gravitačný zákon" 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/Splo%C5%A1ni_gravitacijski_zakon" title="Splošni gravitacijski zakon – Slovenian" lang="sl" hreflang="sl" data-title="Splošni gravitacijski zakon" 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/%DB%8C%D8%A7%D8%B3%D8%A7%DB%8C_%DA%95%D8%A7%DA%A9%DB%8E%D8%B4%D8%A7%D9%86%DB%8C_%DA%AF%DB%95%D8%B1%D8%AF%D9%88%D9%88%D9%86%DB%8C" title="یاسای ڕاکێشانی گەردوونی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="یاسای ڕاکێشانی گەردوونی" 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%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D0%B7%D0%B0%D0%BB%D0%BD%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B5" title="Универзални закон гравитације – Serbian" lang="sr" hreflang="sr" data-title="Универзални закон гравитације" 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/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Newtonov zakon gravitacije" 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/Newtonin_painovoimalaki" title="Newtonin painovoimalaki – Finnish" lang="fi" hreflang="fi" data-title="Newtonin painovoimalaki" 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/Newtons_gravitationslag" title="Newtons gravitationslag – Swedish" lang="sv" hreflang="sv" data-title="Newtons gravitationslag" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Batas_ng_unibersal_na_grabitasyon_ni_Newton" title="Batas ng unibersal na grabitasyon ni Newton – Tagalog" lang="tl" hreflang="tl" data-title="Batas ng unibersal na grabitasyon ni Newton" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%AF%E0%AF%82%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%A9%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%88%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="நியூட்டனின் ஈர்ப்பு விதி – Tamil" lang="ta" hreflang="ta" data-title="நியூட்டனின் ஈர்ப்பு விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%91%D3%A9%D1%82%D0%B5%D0%BD%D0%B4%D3%A9%D0%BD%D1%8C%D1%8F_%D1%82%D0%B0%D1%80%D1%82%D1%8B%D0%BB%D1%83_%D0%BA%D0%B0%D0%BD%D1%83%D0%BD%D1%8B" title="Бөтендөнья тартылу кануны – Tatar" lang="tt" hreflang="tt" data-title="Бөтендөнья тартылу кануны" 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-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A8%E0%B1%8D%E0%B0%AF%E0%B1%82%E0%B0%9F%E0%B0%A8%E0%B1%8D_%E0%B0%B5%E0%B0%BF%E0%B0%B6%E0%B1%8D%E0%B0%B5%E0%B0%97%E0%B1%81%E0%B0%B0%E0%B1%81%E0%B0%A4%E0%B1%8D%E0%B0%B5%E0%B0%BE%E0%B0%95%E0%B0%B0%E0%B1%8D%E0%B0%B7%E0%B0%A3_%E0%B0%B8%E0%B0%BF%E0%B0%A6%E0%B1%8D%E0%B0%A7%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B0%82" title="న్యూటన్ విశ్వగురుత్వాకర్షణ సిద్ధాంతం – Telugu" lang="te" hreflang="te" data-title="న్యూటన్ విశ్వగురుత్వాకర్షణ సిద్ధాంతం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%82%E0%B8%99%E0%B9%89%E0%B8%A1%E0%B8%96%E0%B9%88%E0%B8%A7%E0%B8%87%E0%B8%AA%E0%B8%B2%E0%B8%81%E0%B8%A5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%99%E0%B8%B4%E0%B8%A7%E0%B8%95%E0%B8%B1%E0%B8%99" title="กฎความโน้มถ่วงสากลของนิวตัน – Thai" lang="th" hreflang="th" data-title="กฎความโน้มถ่วงสากลของนิวตัน" 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/Newton%27un_evrensel_k%C3%BCtle%C3%A7ekim_yasas%C4%B1" title="Newton'un evrensel kütleçekim yasası – Turkish" lang="tr" hreflang="tr" data-title="Newton'un evrensel kütleçekim yasası" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%94%D1%83%D0%BD%D0%BD%D0%B5_%D0%BA%D1%8B%D1%81%D1%82%D3%A5%D1%81%D1%8C%D0%BA%D0%BE%D0%BD_%D1%81%D1%8F%D1%80%D1%8B%D1%81%D1%8C_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" title="Дунне кыстӥськон сярысь закон – Udmurt" lang="udm" hreflang="udm" data-title="Дунне кыстӥськон сярысь закон" data-language-autonym="Удмурт" data-language-local-name="Udmurt" class="interlanguage-link-target"><span>Удмурт</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B2%D1%81%D0%B5%D1%81%D0%B2%D1%96%D1%82%D0%BD%D1%8C%D0%BE%D0%B3%D0%BE_%D1%82%D1%8F%D0%B6%D1%96%D0%BD%D0%BD%D1%8F" title="Закон всесвітнього тяжіння – Ukrainian" lang="uk" hreflang="uk" data-title="Закон всесвітнього тяжіння" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D9%86_%DA%A9%D8%A7_%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%B9%D8%A7%D9%84%D9%85%DB%8C_%D8%AB%D9%82%D8%A7%D9%84%D8%AA" title="نیوٹن کا قانون عالمی ثقالت – Urdu" lang="ur" hreflang="ur" data-title="نیوٹن کا قانون عالمی ثقالت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_lu%E1%BA%ADt_v%E1%BA%A1n_v%E1%BA%ADt_h%E1%BA%A5p_d%E1%BA%ABn_c%E1%BB%A7a_Newton" title="Định luật vạn vật hấp dẫn của Newton – Vietnamese" lang="vi" hreflang="vi" data-title="Định luật vạn vật hấp dẫn của Newton" 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 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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 mw-collapsed"><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 href="/wiki/Celestial_mechanics" title="Celestial mechanics">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"><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 class="mw-selflink selflink"><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 href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Edward_Routh" title="Edward Routh">Routh</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a></li> <li><a href="/wiki/Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von 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, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Newton's law of universal gravitation</b> states that every <a href="/wiki/Particle" title="Particle">particle</a> attracts every other particle in the universe with a <a href="/wiki/Force" title="Force">force</a> that is <a href="/wiki/Proportionality_(mathematics)#Direct_proportionality" title="Proportionality (mathematics)">proportional</a> to the product of their masses and <a href="/wiki/Proportionality_(mathematics)#Inverse_proportionality" title="Proportionality (mathematics)">inversely proportional</a> to the square of the distance between their centers. Separated objects attract and are attracted <a href="/wiki/Shell_theorem" title="Shell theorem">as if all their mass were concentrated at their centers</a>. The publication of the law has become known as the "<a href="/wiki/Unification_(physics)#Unification_of_gravity_and_astronomy" class="mw-redirect" title="Unification (physics)">first great unification</a>", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> </p><p>This is a general <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical law</a> derived from <a href="/wiki/Empirical_observation" class="mw-redirect" title="Empirical observation">empirical observations</a> by what <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> called <i><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a></i>.<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> It is a part of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> and was formulated in Newton's work <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a></i> ("the <i>Principia</i>"), first published on 5 July 1687. </p><p>The equation for universal gravitation thus takes the form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48f74b3b4d591ba1996c4d481f74ac3ab7e279d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.338ex; height:5.009ex;" alt="{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}"></span> </p><p>where <i>F</i> is the gravitational force acting between two objects, <i>m<sub>1</sub></i> and <i>m<sub>2</sub></i> are the masses of the objects, <i>r</i> is the distance between the <a href="/wiki/Center_of_mass" title="Center of mass">centers of their masses</a>, and <i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>. </p><p>The first test of Newton's law of gravitation between masses in the laboratory was the <a href="/wiki/Cavendish_experiment" title="Cavendish experiment">Cavendish experiment</a> conducted by the British scientist <a href="/wiki/Henry_Cavendish" title="Henry Cavendish">Henry Cavendish</a> in 1798.<sup id="cite_ref-The_Michell-Cavendish_Experiment_5-0" class="reference"><a href="#cite_note-The_Michell-Cavendish_Experiment-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It took place 111 years after the publication of Newton's <i>Principia</i> and approximately 71 years after his death. </p><p>Newton's law of <a href="/wiki/Gravity" title="Gravity">gravitation</a> resembles <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a> of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square laws</a>, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has charge in place of mass and a different constant. </p><p>Newton's law was later superseded by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, but the universality of the gravitational constant is intact and the law still continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme accuracy, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at small distances (such as <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a>'s orbit around the Sun). </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=Newton%27s_law_of_universal_gravitation&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">Main article: <a href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">History of gravitational theory</a></div> <p>Around 1600, the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a> began to take root. <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> started over with a more fundamental view, developing ideas of matter and action independent of theology. <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> wrote about experimental measurements of falling and rolling objects. <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a>'s <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion"> laws of planetary motion</a> summarized <a href="/wiki/Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a>'s astronomical observations.<sup id="cite_ref-Hesse2005_6-0" class="reference"><a href="#cite_note-Hesse2005-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 132">: 132 </span></sup> </p><p>Around 1666 <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> developed the idea that Kepler's laws must also apply to the orbit of the Moon around the Earth and then to all objects on Earth. The analysis required assuming that the gravitation force acted as if all of the mass of the Earth were concentrated at its center, an unproven conjecture at that time. His calculations of the Moon orbit time was within 16% of the known value. By 1680, new values for the diameter of the Earth improved his orbit time to within 1.6%, but more importantly Newton had found a proof of his earlier conjecture.<sup id="cite_ref-Feather_7-0" class="reference"><a href="#cite_note-Feather-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 201">: 201 </span></sup> </p><p>In 1687 Newton published his <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> which combined his <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion"> laws of motion</a> with new mathematical analysis to explain Kepler's empirical results.<sup id="cite_ref-Hesse2005_6-1" class="reference"><a href="#cite_note-Hesse2005-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 134">: 134 </span></sup> His explanation was in the form of a law of universal gravitation: any two bodies are attracted by a force proportional to their mass and inversely proportional to their separation squared.<sup id="cite_ref-Whittaker_8-0" class="reference"><a href="#cite_note-Whittaker-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 28">: 28 </span></sup> Newton's original formula was: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {Force\,of\,gravity}}\propto {\frac {\rm {mass\,of\,object\,1\,\times \,mass\,of\,object\,2}}{\rm {distance\,from\,centers^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mspace width="thinmathspace" /> <mn>1</mn> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">j</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {Force\,of\,gravity}}\propto {\frac {\rm {mass\,of\,object\,1\,\times \,mass\,of\,object\,2}}{\rm {distance\,from\,centers^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aab6501a8e9379b68a05b789c758f6126e0cab1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:54.088ex; height:5.676ex;" alt="{\displaystyle {\rm {Force\,of\,gravity}}\propto {\frac {\rm {mass\,of\,object\,1\,\times \,mass\,of\,object\,2}}{\rm {distance\,from\,centers^{2}}}}}"></span> </p><p>where the symbol <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 \propto }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∝</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \propto }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3a55007ba2f092d6cafe6d33598e0608b81150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.676ex;" alt="{\displaystyle \propto }"></span> means "is proportional to". To make this into an equal-sided formula or equation, there needed to be a multiplying factor or constant that would give the correct force of gravity no matter the value of the masses or distance between them (the gravitational constant). Newton would need an accurate measure of this constant to prove his inverse-square law. When Newton presented Book 1 of the unpublished text in April 1686 to the <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>, <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> made a claim that Newton had obtained the inverse square law from him, ultimately a frivolous accusation.<sup id="cite_ref-Feather_7-1" class="reference"><a href="#cite_note-Feather-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204">: 204 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Newton's_"causes_hitherto_unknown""><span id="Newton.27s_.22causes_hitherto_unknown.22"></span>Newton's "causes hitherto unknown"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=2" title="Edit section: Newton's "causes hitherto unknown""><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/Action_at_a_distance" title="Action at a distance">Action at a distance</a></div> <p>While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied. In 1692, in his third letter to Bentley, he wrote: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it." </p><p>He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two <a href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">mechanical hypotheses</a> in 1675 and 1717). Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found. And in Newton's 1713 <i><a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a></i> in the second edition of <i>Principia</i>: "I have not yet been able to discover the cause of these properties of gravity from phenomena and I <a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">feign no hypotheses</a>. ... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies."<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Modern_form">Modern form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=3" title="Edit section: Modern form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In modern language, the law states the following: </p> <table style="border-top:1px solid #aaa; border-bottom:1px solid #aaa; margin-bottom:1em"> <tbody><tr> <td style="padding:0.1em 2em 0 0.5em"><span style="white-space:nowrap">Every <a href="/wiki/Point_mass" class="mw-redirect" title="Point mass">point</a> <a href="/wiki/Mass" title="Mass">mass</a> attracts every single other point mass by a <a href="/wiki/Force" title="Force">force</a> acting along</span> the <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> intersecting both points. The force is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of the two masses and <a href="/wiki/Proportionality_(mathematics)#Inverse_proportionality" title="Proportionality (mathematics)">inversely proportional</a> to the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of the distance between them:<sup id="cite_ref-Newton1_10-0" class="reference"><a href="#cite_note-Newton1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td> <td rowspan="2" style="width:99%; vertical-align:top"><figure class="mw-default-size mw-halign-left" typeof="mw:File"><a href="/wiki/File:NewtonsLawOfUniversalGravitation.svg" class="mw-file-description" title="Diagram of two masses attracting one another"><img alt="Diagram of two masses attracting one another" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/400px-NewtonsLawOfUniversalGravitation.svg.png" decoding="async" width="400" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/600px-NewtonsLawOfUniversalGravitation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/800px-NewtonsLawOfUniversalGravitation.svg.png 2x" data-file-width="400" data-file-height="250" /></a><figcaption>Diagram of two masses attracting one another</figcaption></figure> </td></tr> <tr> <td style="padding:0 2em 0.1em 0.5em; white-space:nowrap"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6ee5510ba3c7d6664775c0e76b53e72468303a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.272ex; height:5.009ex;" alt="{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }"></span> where </p> <ul><li><i>F</i> is the force between the masses;</li> <li><i>G</i> is the <a href="/wiki/Newtonian_constant_of_gravitation" class="mw-redirect" title="Newtonian constant of gravitation">Newtonian constant of gravitation</a> (<span class="nowrap"><span data-sort-value="6989667400000000000♠"></span>6.674<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−11</sup> m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup></span>);</li> <li><i>m</i><sub>1</sub> is the first mass;</li> <li><i>m</i><sub>2</sub> is the second mass;</li> <li><i>r</i> is the distance between the centers of the masses.</li></ul> </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravity_Big_G_Measurements_NIST.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/440px-Gravity_Big_G_Measurements_NIST.png" decoding="async" width="440" height="308" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/660px-Gravity_Big_G_Measurements_NIST.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/880px-Gravity_Big_G_Measurements_NIST.png 2x" data-file-width="3000" data-file-height="2100" /></a><figcaption>Error plot showing experimental values for <i>G</i>.</figcaption></figure> <p>Assuming <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>, <i>F</i> is measured in <a href="/wiki/Newton_(units)" class="mw-redirect" title="Newton (units)">newtons</a> (N), <i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> in <a href="/wiki/Kilogram" title="Kilogram">kilograms</a> (kg), <i>r</i> in meters (m), and the constant <i>G</i> is <span class="nowrap"><span data-sort-value="6989667430000000000♠"></span>6.674<span style="margin-left:.25em;">30</span>(15)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−11</sup> m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup></span>.<sup id="cite_ref-physconst-G_11-0" class="reference"><a href="#cite_note-physconst-G-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The value of the constant <i>G</i> was first accurately determined from the results of the <a href="/wiki/Cavendish_experiment" title="Cavendish experiment">Cavendish experiment</a> conducted by the <a href="/wiki/United_Kingdom" title="United Kingdom">British</a> scientist <a href="/wiki/Henry_Cavendish" title="Henry Cavendish">Henry Cavendish</a> in 1798, although Cavendish did not himself calculate a numerical value for <i>G</i>.<sup id="cite_ref-The_Michell-Cavendish_Experiment_5-1" class="reference"><a href="#cite_note-The_Michell-Cavendish_Experiment-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's <i>Principia</i> and 71 years after Newton's death, so none of Newton's calculations could use the value of <i>G</i>; instead he could only calculate a force relative to another force. </p> <div class="mw-heading mw-heading2"><h2 id="Bodies_with_spatial_extent">Bodies with spatial extent</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=4" title="Edit section: Bodies with spatial extent"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Earth-G-force.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/220px-Earth-G-force.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/330px-Earth-G-force.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/440px-Earth-G-force.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Gravitational field strength within the Earth</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravity_field_near_earth.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Gravity_field_near_earth.gif/220px-Gravity_field_near_earth.gif" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/74/Gravity_field_near_earth.gif 1.5x" data-file-width="301" data-file-height="298" /></a><figcaption>Gravity field near the surface of the Earth – an object is shown accelerating toward the surface</figcaption></figure> <p>If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses that constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails <a href="/wiki/Integral" title="Integral">integrating</a> the force (in vector form, see below) over the extents of the two <a href="/wiki/Physical_body" class="mw-redirect" title="Physical body">bodies</a>. </p><p>In this way, it can be shown that an object with a spherically symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.<sup id="cite_ref-Newton1_10-1" class="reference"><a href="#cite_note-Newton1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> (This is not generally true for non-spherically symmetrical bodies.) </p><p>For points <i>inside</i> a spherically symmetric distribution of matter, Newton's <a href="/wiki/Shell_theorem" title="Shell theorem">shell theorem</a> can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance <i>r</i><sub>0</sub> from the center of the mass distribution:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The portion of the mass that is located at radii <span class="nowrap"><i>r</i> < <i>r</i><sub>0</sub></span> causes the same force at the radius <i>r</i><sub>0</sub> as if all of the mass enclosed within a sphere of radius <i>r</i><sub>0</sub> was concentrated at the center of the mass distribution (as noted above).</li> <li>The portion of the mass that is located at radii <span class="nowrap"><i>r</i> > <i>r</i><sub>0</sub></span> exerts <i>no net</i> gravitational force at the radius <i>r</i><sub>0</sub> from the center. That is, the individual gravitational forces exerted on a point at radius <i>r</i><sub>0</sub> by the elements of the mass outside the radius <i>r</i><sub>0</sub> cancel each other.</li></ul> <p>As a consequence, for example, within a shell of uniform thickness and density there is <i>no net</i> gravitational acceleration anywhere within the hollow sphere. </p> <div class="mw-heading mw-heading2"><h2 id="Vector_form">Vector form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=5" title="Edit section: Vector form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravitymacroscopic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/220px-Gravitymacroscopic.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/330px-Gravitymacroscopic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/440px-Gravitymacroscopic.svg.png 2x" data-file-width="296" data-file-height="296" /></a><figcaption>Gravity field surrounding Earth from a macroscopic perspective.</figcaption></figure> <p>Newton's law of universal gravitation can be written as a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> <a href="/wiki/Equation" title="Equation">equation</a> to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{2}}{\hat {\mathbf {r} }}_{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{3}}\mathbf {r} _{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{2}}{\hat {\mathbf {r} }}_{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{3}}\mathbf {r} _{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567704a78e85920f22bd67511b1aa11026326bc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.093ex; height:5.843ex;" alt="{\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{2}}{\hat {\mathbf {r} }}_{21}=-G{m_{1}m_{2} \over {|\mathbf {r} _{21}|}^{3}}\mathbf {r} _{21}}"></span> where </p> <ul><li><b>F</b><sub>21</sub> is the force applied on body 2 exerted by body 1,</li> <li><i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>,</li> <li><i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> are respectively the masses of bodies 1 and 2,</li> <li><b>r</b><sub>21</sub> = <b>r</b><sub>2</sub> − <b>r</b><sub>1</sub> is the <a href="/wiki/Displacement_vector" class="mw-redirect" title="Displacement vector">displacement vector</a> between bodies 1 and 2, and</li> <li><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 {\hat {\mathbf {r} }}_{21}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r_{2}-r_{1}} }{|\mathbf {r_{2}-r_{1}} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> <mo mathvariant="bold">−</mo> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> <mo mathvariant="bold">−</mo> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}_{21}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r_{2}-r_{1}} }{|\mathbf {r_{2}-r_{1}} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9eb55742ba8df83fc77aff3d0caff9b6a87e44e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.254ex; height:5.676ex;" alt="{\displaystyle {\hat {\mathbf {r} }}_{21}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r_{2}-r_{1}} }{|\mathbf {r_{2}-r_{1}} |}}}"></span> is the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> from body 1 to body 2.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <p>It can be seen that the vector form of the equation is the same as the <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> form given earlier, except that <b>F</b> is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that <b>F</b><sub>12</sub> = −<b>F</b><sub>21</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Gravity_field">Gravity field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=6" title="Edit section: Gravity field"><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/Gravitational_field" title="Gravitational field">Gravitational field</a></div> <p>The <b>gravitational field</b> is a <a href="/wiki/Vector_field" title="Vector field">vector field</a> that describes the gravitational force that would be applied on an object in any given point in space, per unit mass. It is actually equal to the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> at that point. </p><p>It is a generalisation of the vector form, which becomes particularly useful if more than two objects are involved (such as a rocket between the Earth and the Moon). For two objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write <b>r</b> instead of <b>r</b><sub>12</sub> and <i>m</i> instead of <i>m</i><sub>2</sub> and define the gravitational field <b>g</b>(<b>r</b>) as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo fence="false" stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf0689fbd05781a129e2df24ef5bd8b7edf2f93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.991ex; height:5.843ex;" alt="{\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }"></span> so that we can write: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef10f4f1d5c39458a54cfc8edcd1e5b9bdda13fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.628ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}"></span> </p><p>This formulation is dependent on the objects causing the field. The field has units of acceleration; in <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a>, this is m/s<sup>2</sup>. </p><p>Gravitational fields are also <a href="/wiki/Conservative_field" class="mw-redirect" title="Conservative field">conservative</a>; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field <i>V</i>(<b>r</b>) such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28ab8b74ac2595cca9eb982767171ceb79ebe90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.436ex; height:2.843ex;" alt="{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}"></span> </p><p>If <i>m</i><sub>1</sub> is a point mass or the mass of a sphere with homogeneous mass distribution, the force field <b>g</b>(<b>r</b>) outside the sphere is isotropic, i.e., depends only on the distance <i>r</i> from the center of the sphere. In that case </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b8de5a197a1b13a6eb7844e692862b3da53ba2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.956ex; height:4.676ex;" alt="{\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}"></span> </p><p>As per <a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law</a>, field in a symmetric body can be found by the mathematical equation: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent"><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></span> <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 \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>enc</mtext> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06858ef1880b87d71ba7d1e5f508944f81d75b2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.167ex; height:2.843ex;" alt="{\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}"></span></span></div> <p>where <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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span> is a closed surface and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{enc}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>enc</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{enc}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36be47bb78936458374b87df625a0d9d28fd9136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.86ex; height:2.509ex;" alt="{\displaystyle M_{\text{enc}}}"></span> is the mass enclosed by the surface. </p><p>Hence, for a hollow sphere of radius <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> and total mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>r</mi> <mo><</mo> <mi>R</mi> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>r</mi> <mo>≥</mo> <mi>R</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a50ff23eb695f4aa44566ad8ae1afba19a2093a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:27.652ex; height:11.176ex;" alt="{\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}"></span> </p><p>For a uniform solid sphere of radius <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> and total mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>r</mi> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>r</mi> <mo><</mo> <mi>R</mi> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>r</mi> <mo>≥</mo> <mi>R</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c883415e05b228c2c5f2fb58d23cd3aff887e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.701ex; height:13.843ex;" alt="{\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Limitations">Limitations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=7" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities <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 \phi /c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi /c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db3f2cd8be8be07f1a16f28254eb406c1bac1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.609ex; height:3.176ex;" alt="{\displaystyle \phi /c^{2}}"></span> and <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 (v/c)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v/c)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b52f8b57e013540544df4327a214f24fad07b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.16ex; height:3.176ex;" alt="{\displaystyle (v/c)^{2}}"></span> are both much less than one, where <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is the <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a>, <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is the velocity of the objects being studied, and <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∼</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e8edb8b51a43bfb6304c275d706233b31b6ba2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.995ex; height:6.676ex;" alt="{\displaystyle {\frac {\phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8},}"></span> </p><p>where <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 r_{\text{orbit}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>orbit</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{orbit}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991691b78b1d896fae06cd7fb7204a2bf82c11c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.759ex; height:2.009ex;" alt="{\displaystyle r_{\text{orbit}}}"></span> is the radius of the Earth's orbit around the Sun. </p><p>In situations where either dimensionless parameter is large, then <a href="/wiki/General_relativity" title="General relativity">general relativity</a> must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity. </p> <div class="mw-heading mw-heading3"><h3 id="Observations_conflicting_with_Newton's_formula"><span id="Observations_conflicting_with_Newton.27s_formula"></span>Observations conflicting with Newton's formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=8" title="Edit section: Observations conflicting with Newton's formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Newton's theory does not fully explain the <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession of the perihelion</a> of the orbits of the planets, especially that of Mercury, which was detected long after the life of Newton.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> There is a 43 <a href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcsecond</a> per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th century.</li> <li>The predicted angular <a href="/wiki/Gravitational_lens" title="Gravitational lens">deflection of light rays by gravity</a> (treated as particles travelling at the expected speed) that is calculated by using Newton's theory is only one-half of the deflection that is observed by astronomers.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="The literature has much controversy on this. (June 2020)">citation needed</span></a></i>]</sup> Calculations using general relativity are in much closer agreement with the astronomical observations.</li> <li>In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey both Newton's law of universal gravitation and general relativity. Astrophysicists, however, explain this marked phenomenon by assuming the presence of large amounts of <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Einstein's_solution"><span id="Einstein.27s_solution"></span>Einstein's solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=9" title="Edit section: Einstein's solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><style data-mw-deduplicate="TemplateStyles:r1247671788">.mw-parser-output .spacetime .sidebar-list-title{background:transparent;border-top:1px solid #aaa;text-align:center}.mw-parser-output .spacetime .sidebar-below{background-color:transparent;border-color:#A2B8BF}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" 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:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks spacetime"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:GPB_circling_earth.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/240px-GPB_circling_earth.jpg" decoding="async" width="240" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/360px-GPB_circling_earth.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/480px-GPB_circling_earth.jpg 2x" data-file-width="1200" data-file-height="900" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li></ul> </div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Spacetime concepts</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Spacetime" title="Spacetime">Spacetime manifold</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</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="color: var(--color-base)">General relativity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction to general relativity</a></li> <li><a href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</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="color: var(--color-base)">Classical gravity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Gravity" title="Gravity">Introduction to gravitation</a></li> <li><a class="mw-selflink selflink">Newton's law of universal gravitation</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="color: var(--color-base)">Relevant mathematics</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Four-vector" title="Four-vector">Four-vector</a></li> <li><a href="/wiki/Derivations_of_the_Lorentz_transformations" title="Derivations of the Lorentz transformations">Derivations of relativity</a></li> <li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Curved_space" title="Curved space">Curved space</a></li> <li><a href="/wiki/Curved_spacetime" title="Curved spacetime">Curved spacetime</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Spacetime_topology" title="Spacetime topology">Spacetime topology</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below"> <div class="hlist"> <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, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Spacetime" title="Category:Spacetime">Category</a></span></li></ul> </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:Spacetime" title="Template:Spacetime"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spacetime" title="Template talk:Spacetime"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spacetime" title="Special:EditPage/Template:Spacetime"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The first two conflicts with observations above were explained by Einstein's theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, in which gravitation is a manifestation of <a href="/wiki/Curved_spacetime" title="Curved spacetime">curved spacetime</a> instead of being due to a force propagated between bodies. In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. In general relativity, the gravitational force is a <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious force</a> resulting from the <a href="/wiki/Curvature_of_spacetime" class="mw-redirect" title="Curvature of spacetime">curvature of spacetime</a>, because the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> of a body in <a href="/wiki/Free_fall" title="Free fall">free fall</a> is due to its <a href="/wiki/World_line" title="World line">world line</a> being a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=10" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In recent years, quests for non-inverse square terms in the law of gravity have been carried out by <a href="/wiki/Neutron_interferometry" class="mw-redirect" title="Neutron interferometry">neutron interferometry</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Solutions">Solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=11" title="Edit section: Solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</a> has been completely solved, as has the restricted <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/N-body_problem" title="N-body problem">n-body problem</a> is an ancient, classical problem<sup id="cite_ref-Leimanis_and_Minorsky_18-0" class="reference"><a href="#cite_note-Leimanis_and_Minorsky-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> of predicting the individual motions of a group of <a href="/wiki/Astronomical_object" title="Astronomical object">celestial objects</a> interacting with each other <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitationally</a>. Solving this problem – from the time of the Greeks and on – has been motivated by the desire to understand the motions of the <a href="/wiki/Sun" title="Sun">Sun</a>, <a href="/wiki/Planet" title="Planet">planets</a> and the visible <a href="/wiki/Star" title="Star">stars</a>. The classical problem can be informally stated as: <i>given the quasi-steady orbital properties</i> (<i>instantaneous position, velocity and time</i>)<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <i>of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times</i>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>In the 20th century, understanding the dynamics of <a href="/wiki/Globular_cluster" title="Globular cluster">globular cluster</a> star systems became an important <i>n</i>-body problem too. The <i>n</i>-body problem in <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is considerably more difficult to solve. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><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/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" 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Einstein frames</a> – different conventions for the metric tensor, in a theory of a dilaton coupled to gravity<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Kepler_orbit" title="Kepler orbit">Kepler orbit</a> – Celestial orbit whose trajectory is a conic section in the orbital plane</li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a> – Thought experiment about gravity</li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> – Laws in physics about force and motion</li> <li><a href="/wiki/Social_gravity" title="Social gravity">Social gravity</a> – Social theory</li> <li><a href="/wiki/Static_forces_and_virtual-particle_exchange" title="Static forces and virtual-particle exchange">Static forces and virtual-particle exchange</a> – Physical interaction in post-classical physics</li></ul> <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=Newton%27s_law_of_universal_gravitation&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:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFFritz_Rohrlich1989" class="citation book cs1">Fritz Rohrlich (25 August 1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3TqA1394OVcC&pg=PA28"><i>From Paradox to Reality: Our Basic Concepts of the Physical World</i></a>. Cambridge University Press. pp. 28ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-37605-1" title="Special:BookSources/978-0-521-37605-1"><bdi>978-0-521-37605-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Paradox+to+Reality%3A+Our+Basic+Concepts+of+the+Physical+World&rft.pages=28ff.&rft.pub=Cambridge+University+Press&rft.date=1989-08-25&rft.isbn=978-0-521-37605-1&rft.au=Fritz+Rohrlich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3TqA1394OVcC%26pg%3DPA28&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMainzer2013" class="citation book cs1">Mainzer, Klaus (2 December 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QekhAAAAQBAJ&pg=PA8"><i>Symmetries of Nature: A Handbook for Philosophy of Nature and Science</i></a>. Walter de Gruyter. pp. 8ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-088693-1" title="Special:BookSources/978-3-11-088693-1"><bdi>978-3-11-088693-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetries+of+Nature%3A+A+Handbook+for+Philosophy+of+Nature+and+Science&rft.pages=8ff.&rft.pub=Walter+de+Gruyter&rft.date=2013-12-02&rft.isbn=978-3-11-088693-1&rft.aulast=Mainzer&rft.aufirst=Klaus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQekhAAAAQBAJ%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" 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 class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science/science-magazines/physics-fundamental-forces-and-synthesis-theory">"Physics: Fundamental Forces and the Synthesis of Theory"</a>. <i>Encyclopedia.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Physics%3A+Fundamental+Forces+and+the+Synthesis+of+Theory&rft.btitle=Encyclopedia.com&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience%2Fscience-magazines%2Fphysics-fundamental-forces-and-synthesis-theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" 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">Isaac Newton: "In [experimental] philosophy particular propositions are inferred from the phenomena and afterwards rendered general by induction": <i><a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Principia</a>', Book 3, </i>General Scholium<i>, at p.392 in Volume 2 of Andrew Motte's English translation published 1729.</i></span> </li> <li id="cite_note-The_Michell-Cavendish_Experiment-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-The_Michell-Cavendish_Experiment_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-The_Michell-Cavendish_Experiment_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Hodges, Laurent. <a rel="nofollow" class="external text" href="http://www.public.iastate.edu/~lhodges/Michell.htm">"The Michell–Cavendish Experiment"</a>. Indiana State University.</span> </li> <li id="cite_note-Hesse2005-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hesse2005_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hesse2005_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHesse2005" class="citation book cs1">Hesse, Mary B. (2005). <i>Forces and fields: the concept of action at a distance in the history of physics</i>. Mineola, New York: Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44240-2" title="Special:BookSources/978-0-486-44240-2"><bdi>978-0-486-44240-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Forces+and+fields%3A+the+concept+of+action+at+a+distance+in+the+history+of+physics&rft.place=Mineola%2C+New+York&rft.pub=Dover&rft.date=2005&rft.isbn=978-0-486-44240-2&rft.aulast=Hesse&rft.aufirst=Mary+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-Feather-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Feather_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Feather_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeather1959" class="citation book cs1">Feather, Norman (1959). <i>An Introduction to the Physics of Mass Length and Time</i>. Edinburgh University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Physics+of+Mass+Length+and+Time&rft.pub=Edinburgh+University+Press&rft.date=1959&rft.aulast=Feather&rft.aufirst=Norman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-Whittaker-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1989" class="citation book cs1">Whittaker, Edmund T. (1989). <i>A history of the theories of aether & electricity. 1: The classical theories</i> (reprinted ed.). New York: Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-26126-3" title="Special:BookSources/978-0-486-26126-3"><bdi>978-0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+the+theories+of+aether+%26+electricity.+1%3A+The+classical+theories&rft.place=New+York&rft.edition=reprinted&rft.pub=Dover&rft.date=1989&rft.isbn=978-0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=Edmund+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Westfall, Richard S. (1978). <i>The Construction of Modern Science: Mechanisms and Mechanics</i>. Cambridge University Press.</span> </li> <li id="cite_note-Newton1-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Newton1_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Newton1_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Proposition 75, Theorem 35: p. 956 – I.Bernard Cohen and Anne Whitman, translators: <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, <i>The Principia</i>: <a href="/wiki/Mathematical_Principles_of_Natural_Philosophy" class="mw-redirect" title="Mathematical Principles of Natural Philosophy">Mathematical Principles of Natural Philosophy</a>. Preceded by <i>A Guide to Newton's Principia</i>, by I.Bernard Cohen. University of California Press 1999 <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/0-520-08816-6" title="Special:BookSources/0-520-08816-6">0-520-08816-6</a> <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/0-520-08817-4" title="Special:BookSources/0-520-08817-4">0-520-08817-4</a></span> </li> <li id="cite_note-physconst-G-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-physconst-G_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?bg">"2022 CODATA Value: Newtonian constant of gravitation"</a>. <i>The NIST Reference on Constants, Units, and Uncertainty</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a>. May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+NIST+Reference+on+Constants%2C+Units%2C+and+Uncertainty&rft.atitle=2022+CODATA+Value%3A+Newtonian+constant+of+gravitation&rft.date=2024-05&rft_id=https%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fbg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/336k/lectures/node109.html">"Rotational Flattening"</a>. <i>farside.ph.utexas.edu</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=farside.ph.utexas.edu&rft.atitle=Rotational+Flattening&rft_id=http%3A%2F%2Ffarside.ph.utexas.edu%2Fteaching%2F336k%2Flectures%2Fnode109.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">The vector difference <b>r</b><sub>2</sub> − <b>r</b><sub>1</sub> points from object 1 to object 2. See Fig. 11–6. of <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_11.html#Ch11-S5">The Feynman Lectures on Physics, Volume I</a>, equation (9.19) of <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_09.html#Ch9-S7">The Feynman Lectures on Physics, Volume I</a> and <a href="/wiki/Euclidean_vector#Addition_and_subtraction" title="Euclidean vector">Euclidean vector#Addition and subtraction</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles W.</a>; <a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip S.</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (1973). <i>Gravitation</i>. New York: W. H. Freeman and Company. p. 1049. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0344-0" title="Special:BookSources/978-0-7167-0344-0"><bdi>978-0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.place=New+York&rft.pages=1049&rft.pub=W.+H.+Freeman+and+Company&rft.date=1973&rft.isbn=978-0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles+W.&rft.au=Thorne%2C+Kip+S.&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Born" title="Max Born">Max Born</a> (1924), <i>Einstein's Theory of Relativity</i> (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and the Earth.)</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreeneGudkov2007" class="citation journal cs1">Greene, Geoffrey L.; Gudkov, Vladimir (2007). "Neutron interferometric method to provide improved constraints on non-Newtonian gravity at the nanometer scale". <i>Physical Review C</i>. <b>75</b> (1): 015501. <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/hep-ph/0608346">hep-ph/0608346</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/2007PhRvC..75a5501G">2007PhRvC..75a5501G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevC.75.015501">10.1103/PhysRevC.75.015501</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39665455">39665455</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+C&rft.atitle=Neutron+interferometric+method+to+provide+improved+constraints+on+non-Newtonian+gravity+at+the+nanometer+scale&rft.volume=75&rft.issue=1&rft.pages=015501&rft.date=2007&rft_id=info%3Aarxiv%2Fhep-ph%2F0608346&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A39665455%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevC.75.015501&rft_id=info%3Abibcode%2F2007PhRvC..75a5501G&rft.aulast=Greene&rft.aufirst=Geoffrey+L.&rft.au=Gudkov%2C+Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary <i>n</i> can be approximated via <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, but in practice such an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the <i>n</i>-body problem may be solved using <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a>, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book <i>Gravitational </i>N<i>-body Simulations</i> listed in the References.</span> </li> <li id="cite_note-Leimanis_and_Minorsky-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Leimanis_and_Minorsky_18-0">^</a></b></span> <span class="reference-text">Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the <i>n</i>-body problem, especially Ms. Kovalevskaya's ~1868–1888, twenty-year complex-variables approach, failure; <b>Section 1: The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics</b> (Chapter 1, <i>the motion of a rigid body about a fixed point</i> (<b>Euler</b> and <b>Poisson</b> <i>equations</i>); Chapter 2, <i>Mathematical Exterior Ballistics</i>), good precursor background to the <i>n</i>-body problem; <b>Section 2: Celestial Mechanics</b> (Chapter 1, <i>The Uniformization of the Three-body Problem</i> (Restricted Three-body Problem); Chapter 2, <i>Capture in the Three-Body Problem</i>; Chapter 3, <i>Generalized n-body Problem</i>).</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><i>Quasi-steady</i> loads refers to the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, a <i>steady-state</i> condition refers to a system's state being invariant to time; otherwise, the first derivatives and all higher derivatives are zero.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">R. M. Rosenberg states the <i>n</i>-body problem similarly (see References): "Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?" Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces <i>first</i> before the motions can be determined.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, 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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:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a class="mw-selflink selflink">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" title="Category:Isaac Newton">Isaac Newton</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div> </div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Theories_of_gravitation" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Theories_of_gravitation" title="Template:Theories of gravitation"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Theories_of_gravitation" title="Template talk:Theories of gravitation"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Theories_of_gravitation" title="Special:EditPage/Template:Theories of gravitation"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Theories_of_gravitation" style="font-size:114%;margin:0 4em"><a href="/wiki/Gravity" title="Gravity">Theories of gravitation</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standard</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Newtonian gravity (NG)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Newton's law of universal gravitation</a></li> <li><a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a></li> <li><a href="/wiki/Poisson%27s_equation#Newtonian_gravity" title="Poisson's equation">Poisson's equation for gravity</a></li> <li><a href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">History of gravitational theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/General_relativity" title="General relativity">General relativity (GR)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics</a></li> <li><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Exact solutions</a></li> <li><a href="/wiki/General_relativity#Further_reading" title="General relativity">Resources</a></li> <li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/Gibbons%E2%80%93Hawking%E2%80%93York_boundary_term" title="Gibbons–Hawking–York boundary term">Gibbons–Hawking–York boundary term</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;"><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Alternatives to<br />general relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Classical theories of gravitation</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Classical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Poincar%C3%A9_gauge_theory&action=edit&redlink=1" class="new" title="Poincaré gauge theory (page does not exist)">Poincaré gauge theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</a></span></li></ul></li> <li><a href="/wiki/Bimetric_gravity" title="Bimetric gravity">Bimetric theories</a></li> <li><a href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</a></li> <li><a href="/wiki/Composite_gravity" title="Composite gravity">Composite gravity</a></li> <li><a href="/wiki/F(R)_gravity" title="F(R) gravity"><i>f</i>(<i>R</i>) gravity</a></li> <li><a href="/wiki/Infinite_derivative_gravity" title="Infinite derivative gravity">Infinite derivative gravity</a></li> <li><a href="/wiki/Massive_gravity" title="Massive gravity">Massive gravity</a></li> <li><a href="/wiki/Modified_Newtonian_dynamics" title="Modified Newtonian dynamics">Modified Newtonian dynamics, MOND</a> <ul><li><span style="font-size:85%;"><a href="/wiki/AQUAL" title="AQUAL">AQUAL</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Tensor%E2%80%93vector%E2%80%93scalar_gravity" title="Tensor–vector–scalar gravity">Tensor–vector–scalar</a></span></li></ul></li> <li><a href="/wiki/Nonsymmetric_gravitational_theory" title="Nonsymmetric gravitational theory">Nonsymmetric gravitation</a></li> <li><a href="/wiki/Scalar%E2%80%93tensor_theory" title="Scalar–tensor theory">Scalar–tensor theories</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke</a></span></li></ul></li> <li><a href="/wiki/Scalar%E2%80%93tensor%E2%80%93vector_gravity" title="Scalar–tensor–vector gravity">Scalar–tensor–vector</a></li> <li><a href="/wiki/Conformal_gravity" title="Conformal gravity">Conformal gravity</a></li> <li><a href="/wiki/Scalar_theories_of_gravitation" title="Scalar theories of gravitation">Scalar theories</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström</a></span></li></ul></li> <li><a href="/wiki/Whitehead%27s_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead</a></li> <li><a href="/wiki/Geometrodynamics" title="Geometrodynamics">Geometrodynamics</a></li> <li><a href="/wiki/Induced_gravity" title="Induced gravity">Induced gravity</a></li> <li><a href="/wiki/Degenerate_Higher-Order_Scalar-Tensor_theories" title="Degenerate Higher-Order Scalar-Tensor theories">Degenerate Higher-Order Scalar-Tensor theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Quantum-mechanical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclidean_quantum_gravity" title="Euclidean quantum gravity">Euclidean quantum gravity</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Spin_foam" title="Spin foam">Spin foam</a></span></li></ul></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">Asymptotic safety in quantum gravity</a></li> <li><a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></li> <li><a href="/wiki/DGP_model" title="DGP model">DGP model</a></li> <li><a href="/wiki/Rainbow_gravity_theory" title="Rainbow gravity theory">Rainbow gravity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric and <br /> quantum-mechanical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative geometry</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Superfluid_vacuum_theory#Logarithmic_BEC_vacuum_theory" title="Superfluid vacuum theory">Logarithmic BEC vacuum</a></span></li></ul></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/M-theory" title="M-theory">M-theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/F-theory" title="F-theory">F-theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_0_string_theory" title="Type 0 string theory">Type 0 string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string theory</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Little_string_theory" title="Little string theory">Little string theory</a></span></li></ul></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></span></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Generalisations / <br /> extensions of GR</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Liouville_gravity" class="mw-redirect" title="Liouville gravity">Liouville gravity</a></li> <li><a href="/wiki/Lovelock_theory_of_gravity" title="Lovelock theory of gravity">Lovelock theory</a></li> <li><a href="/wiki/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</a></li> <li><a href="/wiki/Gauss%E2%80%93Bonnet_gravity" title="Gauss–Bonnet gravity">Gauss–Bonnet gravity</a></li> <li><a href="/wiki/Jackiw%E2%80%93Teitelboim_gravity" title="Jackiw–Teitelboim gravity">Jackiw–Teitelboim gravity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pre-Newtonian<br />theories and<br /><a href="/wiki/Toy_model" title="Toy model">toy models</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aristotelian_physics" title="Aristotelian physics">Aristotelian physics</a></li> <li><a href="/wiki/CGHS_model" title="CGHS model">CGHS model</a></li> <li><a href="/wiki/RST_model" title="RST model">RST model</a></li> <li><a href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">Mechanical explanations</a> <ul><li><span style="font-size:85%;"><a href="/wiki/Le_Sage%27s_theory_of_gravitation" title="Le Sage's theory of gravitation">Fatio–Le Sage</a></span></li> <li><span style="font-size:85%;"><a href="/wiki/Entropic_gravity" title="Entropic gravity">Entropic gravity</a></span></li></ul></li> <li><a href="/wiki/Gravitational_interaction_of_antimatter" title="Gravitational interaction of antimatter">Gravitational interaction of antimatter</a></li> <li><a href="/wiki/Physics_in_the_medieval_Islamic_world" title="Physics in the medieval Islamic world">Physics in the medieval Islamic world</a></li> <li><a href="/wiki/Theory_of_impetus" title="Theory of impetus">Theory of impetus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link 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