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Hamiltonian vector field
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<h2 id="definition">Definition</h2> <p>Suppose that ( M , ω ) {\displaystyle (M,\omega )} is a <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifold</a>. Since the <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a> ω {\displaystyle \omega } is nondegenerate, it sets up a <i>fiberwise-linear</i> <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> </p><p> ω : T M → T ∗ M , {\displaystyle \omega :TM\to T^{*}M,} </p><p>between the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> T M {\displaystyle TM} and the <a href="/facts/Cotangent_bundle/dewBXXxr">cotangent bundle</a> T ∗ M {\displaystyle T^{*}M} , with the inverse </p><p> Ω : T ∗ M → T M , Ω = ω − 1 . {\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.} </p><p>Therefore, <a href="/facts/One-form/NQLbWmtD">one-forms</a> on a symplectic manifold M {\displaystyle M} may be identified with <a href="/facts/Vector_field/ccFNuPPp">vector fields</a> and every <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> H : M → R {\displaystyle H:M\rightarrow \mathbb {R} } determines a unique <a href="/facts/Vector_field/ccFNuPPp">vector field</a> X H {\displaystyle X_{H}} , called the <i>Hamiltonian vector field</i> with the <i>Hamiltonian</i> H {\displaystyle H} , by defining for every vector field Y {\displaystyle Y} on M {\displaystyle M} , </p><p> d H ( Y ) = ω ( X H , Y ) . {\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).} </p><p>Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature. </p> <h2 id="examples">Examples</h2> <p>Suppose that M {\displaystyle M} is a 2 n {\displaystyle 2n} -dimensional symplectic manifold. Then locally, one may choose <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical coordinates</a> ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} on M {\displaystyle M} , in which the symplectic form is expressed as:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> ω = ∑ i d q i ∧ d p i , {\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},} </p><p>where d {\displaystyle \operatorname {d} } denotes the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> and ∧ {\displaystyle \wedge } denotes the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a>. Then the Hamiltonian vector field with Hamiltonian H {\displaystyle H} takes the form:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> X H = ( ∂ H ∂ p i , − ∂ H ∂ q i ) = Ω d H , {\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,} </p><p>where Ω {\displaystyle \Omega } is a 2 n × 2 n {\displaystyle 2n\times 2n} square matrix </p><p> Ω = [ 0 I n − I n 0 ] , {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},} </p><p>and </p><p> d H = [ ∂ H ∂ q i ∂ H ∂ p i ] . {\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.} </p><p>The matrix Ω {\displaystyle \Omega } is frequently denoted with J {\displaystyle \mathbf {J} } . </p><p>Suppose that M = R 2 n {\displaystyle M=\mathbb {R} ^{2n}} is the 2 n {\displaystyle 2n} -dimensional <a href="/facts/Symplectic_vector_space/o95tWKBc">symplectic vector space</a> with (global) canonical coordinates. </p> <ul><li>If H = p i {\displaystyle H=p_{i}} then X H = ∂ / ∂ q i ; {\displaystyle X_{H}=\partial /\partial q^{i};} </li> <li>if H = q i {\displaystyle H=q_{i}} then X H = − ∂ / ∂ p i ; {\displaystyle X_{H}=-\partial /\partial p^{i};} </li> <li>if H = 1 2 ∑ ( p i ) 2 {\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}} then X H = ∑ p i ∂ / ∂ q i ; {\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};} </li> <li>if H = 1 2 ∑ a i j q i q j , a i j = a j i {\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}} then X H = − ∑ a i j q i ∂ / ∂ p j . {\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.} </li></ul> <h2 id="properties">Properties</h2> <ul><li>The assignment f ↦ X f {\displaystyle f\mapsto X_{f}} is <a href="/facts/Linear_map/Q1PBKNVV">linear</a>, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.</li> <li>Suppose that ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} are canonical coordinates on M {\displaystyle M} (see above). Then a curve γ ( t ) = ( q ( t ) , p ( t ) ) {\displaystyle \gamma (t)=(q(t),p(t))} is an <a href="/facts/Integral_curve/wGbVcymt">integral curve</a> of the Hamiltonian vector field X H {\displaystyle X_{H}} <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is a solution of <a href="/facts/Hamilton%2527s_equations/2ReerlNA">Hamilton's equations</a>:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> q ˙ i = ∂ H ∂ p i p ˙ i = − ∂ H ∂ q i . {\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}} </li> <li>The Hamiltonian H {\displaystyle H} is constant along the integral curves, because ⟨ d H , γ ˙ ⟩ = ω ( X H ( γ ) , X H ( γ ) ) = 0 {\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0} . That is, H ( γ ( t ) ) {\displaystyle H(\gamma (t))} is actually independent of t {\displaystyle t} . This property corresponds to the <a href="/facts/Conservation_of_energy/12CdN4Ik">conservation of energy</a> in <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a>.</li> <li>More generally, if two functions F {\displaystyle F} and H {\displaystyle H} have a zero <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson bracket</a> (cf. below), then F {\displaystyle F} is constant along the integral curves of H {\displaystyle H} , and similarly, H {\displaystyle H} is constant along the integral curves of F {\displaystyle F} . This fact is the abstract mathematical principle behind <a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>The <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a> ω {\displaystyle \omega } is preserved by the Hamiltonian flow. Equivalently, the <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> L X H ω = 0 {\displaystyle {\mathcal {L}}_{X_{H}}\omega =0} .</li></ul> <h2 id="poisson-bracket">Poisson bracket</h2> <p>The notion of a Hamiltonian vector field leads to a <a href="/facts/Bilinear_form/gyvzn9ym">skew-symmetric</a> bilinear operation on the differentiable functions on a symplectic manifold M {\displaystyle M} , the <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson bracket</a>, defined by the formula </p><p> { f , g } = ω ( X g , X f ) = d g ( X f ) = L X f g {\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g} </p><p>where L X {\displaystyle {\mathcal {L}}_{X}} denotes the <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> along a vector field X {\displaystyle X} . Moreover, one can check that the following identity holds:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> X { f , g } = − [ X f , X g ] {\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}]} , </p><p>where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f {\displaystyle f} and g {\displaystyle g} . As a consequence (a proof at <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson bracket</a>), the Poisson bracket satisfies the <a href="/facts/Jacobi_identity/C9HoyQ3z">Jacobi identity</a>:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> { { f , g } , h } + { { g , h } , f } + { { h , f } , g } = 0 {\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0} , </p><p>which means that the vector space of differentiable functions on M {\displaystyle M} , endowed with the Poisson bracket, has the structure of a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> over R {\displaystyle \mathbb {R} } , and the assignment f ↦ X f {\displaystyle f\mapsto X_{f}} is a <a href="/facts/Lie_algebra_homomorphism/n2gAUkNq">Lie algebra homomorphism</a>, whose <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">kernel</a> consists of the locally constant functions (constant functions if M {\displaystyle M} is connected). </p> <h2 id="remarks">Remarks</h2> <h2 id="notes">Notes</h2> <h2 id="works-cited">Works cited</h2> <ul><li><a href="/facts/Ralph_Abraham_(mathematician)/3GwuMpup">Abraham, Ralph</a>; <a href="/facts/Jerrold_E._Marsden/I07WtXQe">Marsden, Jerrold E.</a> (1978). <i>Foundations of Mechanics</i>. London: Benjamin-Cummings. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-080530102-1.<i>See section 3.2</i>.</li> <li><a href="/facts/Vladimir_Arnold/IMBhyfMS">Arnol'd, V.I.</a> (1997). <a href="https://archive.org/details/mathematicalmeth0000arno"><i>Mathematical Methods of Classical Mechanics</i></a>. Berlin etc: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-96890-3.</li> <li>Frankel, Theodore (1997). <a href="https://archive.org/details/geometryofphysic0000fran"><i>The Geometry of Physics</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-38753-1.</li> <li>Lee, J. M. (2003), <i>Introduction to Smooth manifolds</i>, Springer Graduate Texts in Mathematics, vol. 218, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95448-1</li> <li><a href="/facts/Dusa_McDuff/NTLKNXzf">McDuff, Dusa</a>; Salamon, D. (1998). <i>Introduction to Symplectic Topology</i>. Oxford Mathematical Monographs. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-850451-9.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Hamiltonian vector field on <a href="/facts/NLab/lpPvRmgo">nLab</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lee 2003, Chapter 18. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lee 2003, Chapter 12. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lee 2003, Chapter 18. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lee 2003, Chapter 18. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>See Lee (2003, Chapter 18) for a very concise statement and proof of Noether's theorem. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lee 2003, Chapter 18. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lee 2003, Chapter 18. - Lee, J. M. (2003), Introduction to Smooth manifolds, Springer Graduate Texts in Mathematics, vol. 218, ISBN 0-387-95448-1 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>