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<title>Mathieu transformation</title>
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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Mathieu transformation</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>Mathieu transformations</b> make up a subgroup of <a href="Canonical_transformation" title="Canonical transformation">canonical transformations</a> preserving the <a href="Differential_form" title="Differential form">differential form</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}p_{i}\delta q_{i}=\sum _{i}P_{i}\delta Q_{i}\,}">
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<annotation encoding="application/x-tex">{\displaystyle \sum _{i}p_{i}\delta q_{i}=\sum _{i}P_{i}\delta Q_{i}\,}</annotation>
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</math></span><img src="./6238dd2418cc1d672ee9deeada836a12a0a55c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.803ex; height:5.509ex;" alt="{\displaystyle \sum _{i}p_{i}\delta q_{i}=\sum _{i}P_{i}\delta Q_{i}\,}" loading="lazy"></span></dd></dl>
<p>The transformation is named after the French mathematician <a href="%C3%89mile_L%C3%A9onard_Mathieu" title="Émile Léonard Mathieu">Émile Léonard Mathieu</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Details">Details</h2></div>
<p>In order to have this <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariance</a>, there should exist at least one <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a> between <span class="mwe-math-element mwe-math-element-inline"><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}}">
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<annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation>
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</math></span><img src="./2752dcbff884354069fe332b8e51eb0a70a531b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><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}}">
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<annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation>
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</math></span><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"></span> <b>only</b> (without any <span class="mwe-math-element mwe-math-element-inline"><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_{i},P_{i}}">
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<annotation encoding="application/x-tex">{\displaystyle p_{i},P_{i}}</annotation>
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</math></span><img src="./c78b10fd5d76c4193c2327c6e1f5ce71575e064b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.384ex; height:2.509ex;" alt="{\displaystyle p_{i},P_{i}}" loading="lazy"></span> involved).
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Omega _{1}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\\&amp;{}\ \ \vdots \\\Omega _{m}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Omega _{1}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\\&amp;{}\ \ \vdots \\\Omega _{m}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\end{aligned}}}</annotation>
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</math></span><img src="./c31890ff300c43749b16ee8636afcaa033345c6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:37.879ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\Omega _{1}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\\&amp;{}\ \ \vdots \\\Omega _{m}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&amp;=0\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<m\leq n}">
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<annotation encoding="application/x-tex">{\displaystyle 1&lt;m\leq n}</annotation>
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</math></span><img src="./ea20887a586e2e459a2e9d96688ea52e1c7de025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.794ex; height:2.343ex;" alt="{\displaystyle 1<m\leq n}" loading="lazy"></span>. When <span class="mwe-math-element mwe-math-element-inline"><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=n}">
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</math></span><img src="./69c9d8e54796e7de7d4738510cc10bc3fc55d48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}" loading="lazy"></span> a Mathieu transformation becomes a <a href="Lagrange_point_transformation" class="mw-redirect" title="Lagrange point transformation">Lagrange point transformation</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Canonical_transformation" title="Canonical transformation">Canonical transformation</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFLanczos,_Cornelius1970" class="citation book cs1"><a href="Lanczos" class="mw-redirect" title="Lanczos">Lanczos</a>, Cornelius (1970). <i>The Variational Principles of Mechanics</i>. Toronto: University of Toronto Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-8020-1743-6</bdi>.</cite></li>
<li><cite id="CITEREFWhittaker,_Edmund" class="citation book cs1"><a href="Edmund_Whittaker" class="mw-redirect" title="Edmund Whittaker">Whittaker</a>, Edmund. <i><a href="A_Treatise_on_the_Analytical_Dynamics_of_Particles_and_Rigid_Bodies" class="mw-redirect" title="A Treatise on the Analytical Dynamics of Particles and Rigid Bodies">A Treatise on the Analytical Dynamics of Particles and Rigid Bodies</a></i>.</cite></li></ul>
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