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mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Définition</span> </button> <ul id="toc-Définition-sublist" class="vector-toc-list"> <li id="toc-Rappels_de_mécanique_hamiltonienne" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rappels_de_mécanique_hamiltonienne"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Rappels de mécanique hamiltonienne</span> </div> </a> <ul id="toc-Rappels_de_mécanique_hamiltonienne-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Critère_d'indépendance_des_constantes_du_mouvement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Critère_d'indépendance_des_constantes_du_mouvement"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Critère d'indépendance des constantes du mouvement</span> </div> </a> <ul id="toc-Critère_d'indépendance_des_constantes_du_mouvement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés_d'un_système_intégrable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriétés_d'un_système_intégrable"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Propriétés d'un système intégrable</span> </div> </a> <ul id="toc-Propriétés_d'un_système_intégrable-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Voir aussi</span> </div> </a> <button 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<div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Système intégrable</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="Aller à un article dans une autre langue. Disponible en 11 langues." > <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-11" 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">11 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B7%D8%A7%D9%85_%D9%82%D8%A7%D8%A8%D9%84_%D9%84%D9%84%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="نطام قابل للتكامل – arabe" lang="ar" hreflang="ar" data-title="نطام قابل للتكامل" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_integrable" title="Sistema integrable – catalan" lang="ca" hreflang="ca" data-title="Sistema integrable" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integrable_system" title="Integrable system – anglais" lang="en" hreflang="en" data-title="Integrable system" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_hamiltoniano_integrable" title="Sistema hamiltoniano integrable – espagnol" lang="es" hreflang="es" data-title="Sistema hamiltoniano integrable" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%A7%D9%85%D8%A7%D9%86%D9%87_%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1" title="سامانه انتگرال‌پذیر – persan" lang="fa" hreflang="fa" data-title="سامانه انتگرال‌پذیر" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%AF%E7%A9%8D%E5%88%86%E7%B3%BB" title="可積分系 – japonais" lang="ja" hreflang="ja" data-title="可積分系" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%81%EB%B6%84%EA%B0%80%EB%8A%A5%EA%B3%84" title="적분가능계 – coréen" lang="ko" hreflang="ko" data-title="적분가능계" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_integr%C3%A1vel" title="Sistema integrável – portugais" lang="pt" hreflang="pt" data-title="Sistema integrável" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D1%87%D0%BD%D0%BE_%D1%80%D0%B5%D1%88%D0%B0%D0%B5%D0%BC%D0%B0%D1%8F_%D0%B7%D0%B0%D0%B4%D0%B0%D1%87%D0%B0" title="Точно решаемая задача – russe" lang="ru" hreflang="ru" data-title="Точно решаемая задача" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk 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hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1957758" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><p>En <a href="/wiki/M%C3%A9canique_hamiltonienne" title="Mécanique hamiltonienne">mécanique hamiltonienne</a>, un <b>système intégrable</b> au sens de <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a> est un système qui possède un nombre suffisant de <a href="/w/index.php?title=Constante_du_mouvement&action=edit&redlink=1" class="new" title="Constante du mouvement (page inexistante)">constantes du mouvement</a> <a href="https://en.wikipedia.org/wiki/Constant_of_motion" class="extiw" title="en:Constant of motion"><span class="indicateur-langue" title="Article en anglais : « Constant of motion »">(en)</span></a> indépendantes. Lorsque le mouvement est borné, la dynamique est alors périodique ou quasi périodique. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=1" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=1" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Rappels_de_mécanique_hamiltonienne"><span id="Rappels_de_m.C3.A9canique_hamiltonienne"></span>Rappels de mécanique hamiltonienne</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=2" title="Modifier la section : Rappels de mécanique hamiltonienne" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=2" title="Modifier le code source de la section : Rappels de mécanique hamiltonienne"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit un système à <i>N</i> <a href="/wiki/Degr%C3%A9_de_libert%C3%A9" class="mw-disambig" title="Degré de liberté">degrés de liberté</a> qui est décrit à l'instant <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> par : </p> <ul><li>les <i>N</i> coordonnées généralisées <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 \{q_{i}(t)\}_{i=1,\dots ,N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{q_{i}(t)\}_{i=1,\dots ,N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58932829f03f3ff213bca4337dd566b9c7a1e535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.01ex; height:3.009ex;" alt="{\displaystyle \{q_{i}(t)\}_{i=1,\dots ,N}}"></span></li> <li>les <i>N</i> moments conjugués <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 \{p_{j}(t)\}_{j=1,\dots ,N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{p_{j}(t)\}_{j=1,\dots ,N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18eec20731a49fe4e39f84aa36899709ad89771f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.363ex; height:3.009ex;" alt="{\displaystyle \{p_{j}(t)\}_{j=1,\dots ,N}}"></span>.</li></ul> <p>À chaque instant, les <i>2N</i> coordonnées <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 (q_{i}(t),p_{j}(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q_{i}(t),p_{j}(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1674b832deffe946f532e02c4cbfac2008d0a7e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.057ex; height:3.009ex;" alt="{\displaystyle (q_{i}(t),p_{j}(t))}"></span> définissent <i>un point</i> dans l'<a href="/wiki/Espace_des_phases" title="Espace des phases">espace des phases</a> Γ = ℝ<sup>2<i>N</i></sup>. L'évolution dynamique du système sous le flot hamiltonien se traduit par une courbe continue appelée <i>orbite</i> dans cet espace des phases. Un système hamiltonien <i>invariant par translation dans le temps</i> satisfait toujours à la <a href="/wiki/Conservation_de_l%27%C3%A9nergie" title="Conservation de l'énergie">conservation de l'énergie</a> : </p> <center><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 \quad H(q_{i}(t),p_{j}(t))\ =\ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad H(q_{i}(t),p_{j}(t))\ =\ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52be66a8fdb6ffd23bc573708408255aa883b8cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.479ex; height:3.009ex;" alt="{\displaystyle \quad H(q_{i}(t),p_{j}(t))\ =\ E}"></span></center> <p>de telle sorte que sa dynamique est en fait restreinte à une hypersurface <i>S</i><sub><i>E</i></sub>⊂Γ à 2<i>N</i>-1 dimensions. </p> <div class="mw-heading mw-heading3"><h3 id="Critère_d'indépendance_des_constantes_du_mouvement"><span id="Crit.C3.A8re_d.27ind.C3.A9pendance_des_constantes_du_mouvement"></span>Critère d'indépendance des constantes du mouvement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=3" title="Modifier la section : Critère d'indépendance des constantes du mouvement" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=3" title="Modifier le code source de la section : Critère d'indépendance des constantes du mouvement"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considérons un système hamiltonien invariant par translation dans le temps qui possède <i>N</i> constantes du mouvement en comprenant l'énergie. Soient {<i>F</i><sub><i>i</i></sub>}<sub>{<i>i</i>=1,…,<i>N</i>}</sub> ces <i>N</i> constantes du mouvement. Pour que le système soit intégrable au sens de Liouville, ces constantes doivent être <b>en involution</b>, c’est-à-dire que leurs <a href="/wiki/Crochet_de_Poisson" title="Crochet de Poisson">crochets de Poisson</a> vérifient : </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \ (i,j)\ ,\qquad \left\{F_{i},\ F_{j}\right\}\ =\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <mrow> <mo>{</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ (i,j)\ ,\qquad \left\{F_{i},\ F_{j}\right\}\ =\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42737b2a15bed91a5c4eb101e53cb2dc465d77dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.184ex; height:3.009ex;" alt="{\displaystyle \forall \ (i,j)\ ,\qquad \left\{F_{i},\ F_{j}\right\}\ =\ 0}"></span> </p> </center> <div class="mw-heading mw-heading3"><h3 id="Propriétés_d'un_système_intégrable"><span id="Propri.C3.A9t.C3.A9s_d.27un_syst.C3.A8me_int.C3.A9grable"></span>Propriétés d'un système intégrable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=4" title="Modifier la section : Propriétés d'un système intégrable" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=4" title="Modifier le code source de la section : Propriétés d'un système intégrable"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lorsque le mouvement est borné, on peut trouver une <a href="/wiki/Transformation_canonique" title="Transformation canonique">transformation canonique</a> des <i>2N</i> variables originales (<i>q</i><sub><i>i</i></sub>,<i>p</i><sub><i>j</i></sub>) vers <i>2N</i> nouvelles variables constantes du mouvement « actions-angles » (<i>I</i><sub><i>i</i></sub>,<i>θ</i><sup><i>j</i></sup>) l'Hamiltonien ne dépend plus que des <i>N</i> variables d'action : <i>I</i><sub><i>i</i></sub>. C'est le <a href="/wiki/Th%C3%A9or%C3%A8me_d%27Arnold-Liouville-Mineur" title="Théorème d'Arnold-Liouville-Mineur">théorème d'Arnold-Liouville-Mineur</a>. Dans les coordonnées action-angle, on a : </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(q_{i}(t),p_{j}(t))\ =\ {\tilde {H}}(I,t)\ =\ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(q_{i}(t),p_{j}(t))\ =\ {\tilde {H}}(I,t)\ =\ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6732e290d0c36a3e5544d5f5895d5a00d4806e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.334ex; height:3.343ex;" alt="{\displaystyle H(q_{i}(t),p_{j}(t))\ =\ {\tilde {H}}(I,t)\ =\ E}"></span> </p> </center> <p>Dans ce cas, les équations canoniques de Hamilton pour les actions deviennent : </p> <center><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 {\frac {dI_{i}}{dt}}\ =\ -\ {\frac {\partial {\tilde {H}}}{\partial \theta ^{i}}}\ =\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dI_{i}}{dt}}\ =\ -\ {\frac {\partial {\tilde {H}}}{\partial \theta ^{i}}}\ =\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86fce018ae84df9337083d70e2c8bbe6b1d349d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.196ex; height:6.009ex;" alt="{\displaystyle {\frac {dI_{i}}{dt}}\ =\ -\ {\frac {\partial {\tilde {H}}}{\partial \theta ^{i}}}\ =\ 0}"></span></center> <p>donc les actions <i>I</i> sont toutes des constantes. Par ailleurs, on a également les équations canoniques de Hamilton pour les angles : </p> <center><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 {\frac {d\theta ^{j}(t)}{dt}}\ =\ {\frac {\partial {\tilde {H}}}{\partial I_{j}}}\ =\ \omega ^{j}(I_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\theta ^{j}(t)}{dt}}\ =\ {\frac {\partial {\tilde {H}}}{\partial I_{j}}}\ =\ \omega ^{j}(I_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569cc511a236bdae9c4a0903b8d8406232d22b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.426ex; height:6.676ex;" alt="{\displaystyle {\frac {d\theta ^{j}(t)}{dt}}\ =\ {\frac {\partial {\tilde {H}}}{\partial I_{j}}}\ =\ \omega ^{j}(I_{i})}"></span></center> <p>Les vitesses angulaires sont donc <i>indépendantes du temps</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>, de telle sorte que les angles augmentent linéairement avec le temps, et le mouvement est alors <i>quasi périodique</i> : </p> <center> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{j}(t)\ =\ \omega ^{j}(I_{i})\ t\ +\ \theta ^{j}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>t</mi> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{j}(t)\ =\ \omega ^{j}(I_{i})\ t\ +\ \theta ^{j}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cc9062edc2aafe5e9cec8b0f547fe4873299a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.29ex; height:3.176ex;" alt="{\displaystyle \theta ^{j}(t)\ =\ \omega ^{j}(I_{i})\ t\ +\ \theta ^{j}(0)}"></span> </p> </center> <p>Ainsi, lorsque le mouvement est borné, la dynamique d'un système intégrable est-elle restreinte à un <i>tore invariant</i> <i>T</i><sub><i>N</i></sub>⊂Γ à <i>N</i> dimensions dans l'espace des phases, au lieu d'explorer toute l'hypersurface d'énergie <i>S</i><sub><i>E</i></sub> a priori accessible. Ce tore invariant est caractérisé par la valeurs des <i>N</i> actions, et l'espace de phases Γ est ainsi localement <i>feuilleté</i> par ces tores invariants, correspondants aux différentes valeurs possibles des actions. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=5" title="Modifier la section : Note" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=5" title="Modifier le code source de la section : Note"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Par contre, les vitesses angulaires dépendent en général des valeurs des actions.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=6" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=6" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=7" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=7" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Conditions_d%27int%C3%A9grabilit%C3%A9_pour_les_syst%C3%A8mes_diff%C3%A9rentiels&action=edit&redlink=1" class="new" title="Conditions d'intégrabilité pour les systèmes différentiels (page inexistante)">Conditions d'intégrabilité pour les systèmes différentiels</a> <a href="https://en.wikipedia.org/wiki/Integrability_conditions_for_differential_systems" class="extiw" title="en:Integrability conditions for differential systems"><span class="indicateur-langue" title="Article en anglais : « Integrability conditions for differential systems »">(en)</span></a></li> <li><a href="/wiki/Hypoth%C3%A8se_ergodique" title="Hypothèse ergodique">Hypothèse ergodique</a></li> <li><a href="/wiki/Int%C3%A9grateur_symplectique" title="Intégrateur symplectique">Intégrateur symplectique</a></li> <li><a href="/wiki/M%C3%A9canique_hamiltonienne" title="Mécanique hamiltonienne">Mécanique hamiltonienne</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_de_Frobenius_(g%C3%A9om%C3%A9trie_diff%C3%A9rentielle)" title="Théorème de Frobenius (géométrie différentielle)">Théorème de Frobenius (géométrie différentielle)</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_KAM" title="Théorème KAM">Théorème KAM</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&veaction=edit&section=8" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Syst%C3%A8me_int%C3%A9grable&action=edit&section=8" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="LandauLifchits"><span class="ouvrage" id="Lev_LandauEvgueni_Lifchits"><a href="/wiki/Lev_Landau" title="Lev Landau">Lev Landau</a> et <a href="/wiki/Evgueni_Lifchits" title="Evgueni Lifchits">Evgueni Lifchits</a>, <cite class="italique">Physique théorique</cite>, <abbr class="abbr" title="tome">t.</abbr> 1 : <i>Mécanique</i> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Physique_th%C3%A9orique_(Landau_et_Lifchitz)" title="Référence:Physique théorique (Landau et Lifchitz)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physique+th%C3%A9orique&rft.aulast=Landau&rft.aufirst=Lev&rft.au=Evgueni+Lifchits&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASyst%C3%A8me+int%C3%A9grable"></span></span></span></li> <li><a href="/wiki/Thomas_Kibble" title="Thomas Kibble">T. W. B. Kibble</a> et F.H. Berkshire ; <i>Classical Mechanics</i>, <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <abbr class="abbr" title="Quatrième">4<sup>e</sup></abbr> éd., 1997 <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/058225972X" title="Spécial:Ouvrages de référence/058225972X"><span class="nowrap">058225972X</span></a>)</small><div style="margin-left:2em; line-height:1.5;">Un excellent cours d'introduction à la mécanique, des fondements Newtoniens jusqu'au formalismes plus avancés de Lagrange et de Hamilton. Kibble est professeur émérite de physique théorique de l'Imperial College de Londres. Pour cette <abbr class="abbr" title="Quatrième">4<sup>e</sup></abbr> édition (avec un coauteur), deux chapitres d'introduction aux idées de la théorie du chaos ont été inclus. Niveau : à partir du premier cycle universitaire. (N.B. : Il a existé une traduction française de l'édition précédente, publiée en son temps par Dunod.)</div></li> <li><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Herbert Goldstein</a>, Charles P. Poole et John L. Safko, <i>Classical mechanics</i>, <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <abbr class="abbr" title="Troisième">3<sup>e</sup></abbr> éd., 2001<div style="margin-left:2em; line-height:1.5;">Cet ouvrage de Goldstein est une référence absolue concernant les aspects théoriques modernes de la mécanique - formulations Lagrangienne et Hamiltonienne. Cette troisième édition, réalisée en collaboration, est complétée par un chapitre (chap. 10) sur les développements récents de la théorie du chaos. Le chapitre 3, consacré au problème à 3 corps, a été également partiellement remanié. Niveau second cycle universitaire. (Il a existé autrefois une traduction française d'une édition précédente.)</div></li> <li><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Vladimir I. Arnold</a>, <i>Mathematical Methods of Classical Mechanics</i>, Springer-Verlag, <abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> éd., 1989 <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-387-96890-3" title="Spécial:Ouvrages de référence/0-387-96890-3"><span class="nowrap">0-387-96890-3</span></a>)</small><div style="margin-left:2em; line-height:1.5;">Une synthèse de l'état de l'art en mécanique analytique (formalismes Lagrangien et Hamiltonien) avec l'accent mis sur l'interprétation géométrique de ces formalismes, par l'un des plus brillants mathématiciens du domaine. À partir du second cycle universitaire.</div></li> <li>Vladimir I. Arnold, V. V. Kozlov et A. I. Neishtadt, <i>Mathematical Aspects of Classical and Celestial Mechanics</i>, Encyclopaedia of Mathematical Sciences, Springer-Verlag, <abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> éd., 1993</li> <li>Vladimir I. Arnold et André Avez, <i>Ergodic Problems of Classical Mechanics</i>, Advanced Book Classics, Pearson Addison Wesley, 1989 <small>(<a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <span class="noarchive"><a rel="nofollow" class="external text" href="https://www.amazon.fr/s/?url=search-alias&field-keywords=0201094061&lang=fr">0201094061</a></span>)</small></li> <li><a href="/wiki/Ralph_Abraham_(math%C3%A9maticien)" title="Ralph Abraham (mathématicien)">R. Abraham</a> et <a href="/wiki/Jerrold_Marsden" title="Jerrold Marsden">J. E. Marsden</a>, <i>Foundations of mechanics</i>, the Benjamin/Cummings Publishing Company, <abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> éd., 1978<div style="margin-left:2em; line-height:1.5;">Un livre imposant qui présente un exposé axiomatique rigoureux de la mécanique «  à la Bourbaki », à réserver aux esprits matheux. 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