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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Lagrange bracket</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><b>Lagrange brackets</b> are certain expressions closely related to <a href="Poisson_bracket" title="Poisson bracket">Poisson brackets</a> that were introduced by <a href="Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> in 1808–1810 for the purposes of mathematical formulation of <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, but unlike the Poisson brackets, have fallen out of use.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Suppose that (<i>q</i><sub>1</sub>, ..., <i>q</i><sub><i>n</i></sub>, <i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>n</i></sub>) is a system of <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> on a <a href="Phase_space" title="Phase space">phase space</a>. If each of them is expressed as a function of two variables, <i>u</i> and <i>v</i>, then the Lagrange bracket of <i>u</i> and <i>v</i> is defined by the formula

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<annotation encoding="application/x-tex">{\displaystyle [u,v]_{p,q}=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right).}</annotation>

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</math></span><img src="./e05cf21cdb0f56601481458108913aab286f8349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.422ex; height:6.843ex;" alt="{\displaystyle [u,v]_{p,q}=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ul><li>Lagrange brackets do not depend on the system of <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> (<i>q</i>, <i>p</i>). If (<i>Q</i>,<i>P</i>) = (<i>Q</i><sub>1</sub>, ..., <i>Q</i><sub><i>n</i></sub>, <i>P</i><sub>1</sub>, ..., <i>P</i><sub><i>n</i></sub>) is another system of canonical coordinates, so that <span class="mwe-math-element mwe-math-element-block"><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 Q=Q(q,p),P=P(q,p)}">

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<annotation encoding="application/x-tex">{\displaystyle Q=Q(q,p),P=P(q,p)}</annotation>

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</math></span></span> is a <a href="Canonical_transformation" title="Canonical transformation">canonical transformation</a>, then the Lagrange bracket is an invariant of the transformation, in the sense that <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [u,v]_{q,p}=[u,v]_{Q,P}}">

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<annotation encoding="application/x-tex">{\displaystyle [u,v]_{q,p}=[u,v]_{Q,P}}</annotation>

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</math></span></span> Therefore, the subscripts indicating the canonical coordinates are often omitted.</li>

<li>If <i>Ω</i> is the <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> on the <i>2n</i>-dimensional phase space <i>W</i> and <i>u</i><sub><i>1</i></sub>,...,<i>u</i><sub><i>2n</i></sub> form a system of coordinates on <i>W</i>, the symplectic form can be written as <span class="mwe-math-element mwe-math-element-block"><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 \Omega ={\frac {1}{2}}\Omega _{ij}du^{i}\wedge du^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {1}{2}}\Omega _{ij}du^{i}\wedge du^{j}}</annotation>

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</math></span></span> where the matrix <span class="mwe-math-element mwe-math-element-block"><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 \Omega _{ij}=[u_{i},u_{j}]_{p,q},\quad 1\leq i,j\leq 2n}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{ij}=[u_{i},u_{j}]_{p,q},\quad 1\leq i,j\leq 2n}</annotation>

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</math></span></span> represents the components of <span class="texhtml">Ω</span>, viewed as a <a href="Tensor" title="Tensor">tensor</a>, in the coordinates <i>u</i>. This matrix is the <a href="Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> of the matrix formed by the Poisson brackets <span class="mwe-math-element mwe-math-element-block"><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 \left(\Omega ^{-1}\right)_{ij}=\{u_{i},u_{j}\},\quad 1\leq i,j\leq 2n}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\Omega ^{-1}\right)_{ij}=\{u_{i},u_{j}\},\quad 1\leq i,j\leq 2n}</annotation>

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</math></span></span> of the coordinates <i>u</i>.</li>

<li>As a corollary of the preceding properties, coordinates (<i>Q</i><sub>1</sub>, ..., <i>Q</i><sub><i>n</i></sub>, <i>P</i><sub>1</sub>, ..., <i>P</i><sub><i>n</i></sub>) on a phase space are canonical if and only if the Lagrange brackets between them have the form <span class="mwe-math-element mwe-math-element-block"><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 [Q_{i},Q_{j}]_{p,q}=0,\quad [P_{i},P_{j}]_{p,q}=0,\quad [Q_{i},P_{j}]_{p,q}=-[P_{j},Q_{i}]_{p,q}=\delta _{ij}.}">

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<annotation encoding="application/x-tex">{\displaystyle [Q_{i},Q_{j}]_{p,q}=0,\quad [P_{i},P_{j}]_{p,q}=0,\quad [Q_{i},P_{j}]_{p,q}=-[P_{j},Q_{i}]_{p,q}=\delta _{ij}.}</annotation>

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</math></span></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Lagrange_matrix_in_canonical_transformations">Lagrange matrix in canonical transformations</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Canonical_transformation" title="Canonical transformation">Canonical transformation</a></div>

<p>The concept of Lagrange brackets can be expanded to that of matrices by defining the Lagrange matrix.

</p><p>Consider the following canonical transformation:<span class="mwe-math-element mwe-math-element-block"><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 \eta ={\begin{bmatrix}q_{1}\\\vdots \\q_{N}\\p_{1}\\\vdots \\p_{N}\\\end{bmatrix}}\quad \rightarrow \quad \varepsilon ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{N}\\P_{1}\\\vdots \\P_{N}\\\end{bmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta ={\begin{bmatrix}q_{1}\\\vdots \\q_{N}\\p_{1}\\\vdots \\p_{N}\\\end{bmatrix}}\quad \rightarrow \quad \varepsilon ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{N}\\P_{1}\\\vdots \\P_{N}\\\end{bmatrix}}}</annotation>

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</p><p>Defining <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="{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}">

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<annotation encoding="application/x-tex">{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}</annotation>

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</math></span><img src="./049891c97cd0b9c16a5383ce84ee604db8b234a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.404ex; height:4.843ex;" alt="{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}" loading="lazy"></span>, the Lagrange matrix is defined as <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="{\textstyle {\mathcal {L}}(\eta )=M^{T}JM}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}(\eta )=M^{T}JM}</annotation>

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</math></span><img src="./026acd869d014d3319079840019ce6881ce416e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.483ex; height:3.009ex;" alt="{\textstyle {\mathcal {L}}(\eta )=M^{T}JM}" loading="lazy"></span>, 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 J}">

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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

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</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:

</p><p><span class="mwe-math-element mwe-math-element-block"><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 {\mathcal {L}}_{ij}(\eta )=[M^{T}JM]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{j}}}-{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{k}}{\partial \eta _{j}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial Q_{k}}{\partial \eta _{i}}}{\frac {\partial P_{k}}{\partial \eta _{j}}}-{\frac {\partial P_{k}}{\partial \eta _{i}}}{\frac {\partial Q_{k}}{\partial \eta _{j}}}\right)=[\eta _{i},\eta _{j}]_{\varepsilon }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{ij}(\eta )=[M^{T}JM]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{j}}}-{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{k}}{\partial \eta _{j}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial Q_{k}}{\partial \eta _{i}}}{\frac {\partial P_{k}}{\partial \eta _{j}}}-{\frac {\partial P_{k}}{\partial \eta _{i}}}{\frac {\partial Q_{k}}{\partial \eta _{j}}}\right)=[\eta _{i},\eta _{j}]_{\varepsilon }}</annotation>

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</p><p>The Lagrange matrix satisfies the following known properties:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {L}}^{T}&=-{\mathcal {L}}\\|{\mathcal {L}}|&={|M|^{2}}\\{\mathcal {L}}^{-1}(\eta )&=-M^{-1}J(M^{-1})^{T}=-{\mathcal {P}}(\eta )\\\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}^{T}&=-{\mathcal {L}}\\|{\mathcal {L}}|&={|M|^{2}}\\{\mathcal {L}}^{-1}(\eta )&=-M^{-1}J(M^{-1})^{T}=-{\mathcal {P}}(\eta )\\\end{aligned}}}</annotation>

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</math></span></span>where the <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="{\textstyle {\mathcal {P}}(\eta )}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {P}}(\eta )}</annotation>

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</math></span><img src="./c6cd7f971f55e0afca0910b6e835a68054e742e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.682ex; height:2.843ex;" alt="{\textstyle {\mathcal {P}}(\eta )}" loading="lazy"></span> is known as a Poisson matrix and whose elements correspond to <a href="Poisson_bracket" title="Poisson bracket">Poisson brackets</a>. The last identity can also be stated as the following:<span class="mwe-math-element mwe-math-element-block"><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 \sum _{k=1}^{2N}\{\eta _{i},\eta _{k}\}[\eta _{k},\eta _{j}]=-\delta _{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{2N}\{\eta _{i},\eta _{k}\}[\eta _{k},\eta _{j}]=-\delta _{ij}}</annotation>

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</math></span></span>Note that the summation here involves generalized coordinates as well as generalized momentum.

</p><p>The invariance of Lagrange bracket can be expressed as: <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="{\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij}}">

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<annotation encoding="application/x-tex">{\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij}}</annotation>

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</math></span><img src="./b5acafa81fa45d4ddf2bb1310d47fd8f8ca64938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.718ex; height:3.009ex;" alt="{\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij}}" loading="lazy"></span>, which directly leads to the symplectic condition: <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="{\textstyle M^{T}JM=J}">

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<annotation encoding="application/x-tex">{\textstyle M^{T}JM=J}</annotation>

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</math></span><img src="./13b136a70a8c676028d5038dd14fbf8cf7c4d4b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.371ex; height:2.509ex;" alt="{\textstyle M^{T}JM=J}" loading="lazy"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li>

<li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGiacaglia1972" class="citation book cs1">Giacaglia, Giorgio E. O. (1972). <i>Perturbation methods in non-linear systems</i>. Applied mathematical sciences. New York Heidelberg: Springer. pp. <span class="nowrap">8–</span>9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-90054-2</bdi>.</cite></span>

</li>

</ol></div></div>

<ul><li><a href="Cornelius_Lanczos" title="Cornelius Lanczos">Cornelius Lanczos</a>, <i>The Variational Principles of Mechanics</i>, Dover (1986), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-65067-7</bdi>.</li>

<li>Iglesias, Patrick, <i>Les origines du calcul symplectique chez Lagrange</i> [The origins of symplectic calculus in Lagrange's work], L'Enseign. Math. (2) 44 (1998), no. 3-4, 257–277. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1659212">1659212</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Lagrange_bracket"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LagrangeBracket.html">"Lagrange bracket"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><cite id="CITEREFA.P._Soldatov2001" class="citation cs2">A.P. Soldatov (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lagrange_bracket">"Lagrange bracket"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul></div><div><div class="zim-footer">

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