Hamiltonian vector field

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.^[1]

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g} on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g}.

Suppose that ${\displaystyle (M,\omega )}${\displaystyle (M,\omega )} is a symplectic manifold. Since the symplectic form ${\displaystyle \omega }${\displaystyle \omega } is nondegenerate, it sets up a fiberwise-linear isomorphism

$${\displaystyle \omega :TM\to T^{*}M,}$${\displaystyle \omega :TM\to T^{*}M,}

between the tangent bundle ${\displaystyle TM}${\displaystyle TM} and the cotangent bundle ${\displaystyle T^{*}M}${\displaystyle T^{*}M}, with the inverse

$${\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}$${\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}

Therefore, one-forms on a symplectic manifold ${\displaystyle M}${\displaystyle M} may be identified with vector fields and every differentiable function ${\displaystyle H:M\rightarrow \mathbb {R} }${\displaystyle H:M\rightarrow \mathbb {R} } determines a unique vector field ${\displaystyle X_{H}}${\displaystyle X_{H}}, called the Hamiltonian vector field with the Hamiltonian ${\displaystyle H}${\displaystyle H}, by defining for every vector field ${\displaystyle Y}${\displaystyle Y} on ${\displaystyle M}${\displaystyle M},

$${\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}$${\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}Or more succinctly, ${\displaystyle \iota _{X_{H}}\omega =dH}${\displaystyle \iota _{X_{H}}\omega =dH}.

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.

Suppose that ${\displaystyle M}${\displaystyle M} is a ${\displaystyle 2n}${\displaystyle 2n}-dimensional symplectic manifold. Then locally, one may choose canonical coordinates ${\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}${\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} on ${\displaystyle M}${\displaystyle M}, in which the symplectic form is expressed as:^[2] ${\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}${\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}

where ${\displaystyle \operatorname {d} }${\displaystyle \operatorname {d} } denotes the exterior derivative and ${\displaystyle \wedge }${\displaystyle \wedge } denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian ${\displaystyle H}${\displaystyle H} takes the form:^[1] ${\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega ,円\mathrm {d} H,}${\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega ,円\mathrm {d} H,}

where ${\displaystyle \Omega }${\displaystyle \Omega } is a ${\displaystyle 2n\times 2n}${\displaystyle 2n\times 2n} square matrix

$${\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}$${\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}

and

$${\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}$${\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}

The matrix ${\displaystyle \Omega }${\displaystyle \Omega } is frequently denoted with ${\displaystyle \mathbf {J} }${\displaystyle \mathbf {J} }.

Suppose that ${\displaystyle M=\mathbb {R} ^{2n}}${\displaystyle M=\mathbb {R} ^{2n}} is the ${\displaystyle 2n}${\displaystyle 2n}-dimensional symplectic vector space with (global) canonical coordinates.

• If ${\displaystyle H=p_{i}}${\displaystyle H=p_{i}} then ${\displaystyle X_{H}=\partial /\partial q^{i};}${\displaystyle X_{H}=\partial /\partial q^{i};}
• if ${\displaystyle H=q_{i}}${\displaystyle H=q_{i}} then ${\displaystyle X_{H}=-\partial /\partial p^{i};}${\displaystyle X_{H}=-\partial /\partial p^{i};}
• if ${\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}}${\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}} then ${\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}${\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}
• if ${\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}}${\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}} then ${\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}${\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}

• The assignment ${\displaystyle f\mapsto X_{f}}${\displaystyle f\mapsto X_{f}} is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that ${\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}${\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} are canonical coordinates on ${\displaystyle M}${\displaystyle M} (see above). Then a curve ${\displaystyle \gamma (t)=(q(t),p(t))}${\displaystyle \gamma (t)=(q(t),p(t))} is an integral curve of the Hamiltonian vector field ${\displaystyle X_{H}}${\displaystyle X_{H}} if and only if it is a solution of Hamilton's equations:^[1] $${\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}$${\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}
• The Hamiltonian ${\displaystyle H}${\displaystyle H} is constant along the integral curves, because ${\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}${\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}. That is, ${\displaystyle H(\gamma (t))}${\displaystyle H(\gamma (t))} is actually independent of ${\displaystyle t}${\displaystyle t}. This property corresponds to the conservation of energy in Hamiltonian mechanics.
• More generally, if two functions ${\displaystyle F}${\displaystyle F} and ${\displaystyle H}${\displaystyle H} have a zero Poisson bracket (cf. below), then ${\displaystyle F}${\displaystyle F} is constant along the integral curves of ${\displaystyle H}${\displaystyle H}, and similarly, ${\displaystyle H}${\displaystyle H} is constant along the integral curves of ${\displaystyle F}${\displaystyle F}. This fact is the abstract mathematical principle behind Noether's theorem.^{[nb 1]}
• The symplectic form ${\displaystyle \omega }${\displaystyle \omega } is preserved by the Hamiltonian flow. Equivalently, the Lie derivative ${\displaystyle {\mathcal {L}}_{X_{H}}\omega =0}${\displaystyle {\mathcal {L}}_{X_{H}}\omega =0}.

The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold ${\displaystyle M}${\displaystyle M}, the Poisson bracket , defined by the formula

$${\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}$${\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}

where ${\displaystyle {\mathcal {L}}_{X}}${\displaystyle {\mathcal {L}}_{X}} denotes the Lie derivative along a vector field ${\displaystyle X}${\displaystyle X}. Moreover, one can check that the following identity holds:^[1] ${\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}]}${\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}]},

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g}. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:^[1] ${\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0}${\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0},

which means that the vector space of differentiable functions on ${\displaystyle M}${\displaystyle M}, endowed with the Poisson bracket, has the structure of a Lie algebra over ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} }, and the assignment ${\displaystyle f\mapsto X_{f}}${\displaystyle f\mapsto X_{f}} is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if ${\displaystyle M}${\displaystyle M} is connected).

• Hamiltonian vector field on nLab

• Hamiltonian mechanics
• Symplectic geometry
• William Rowan Hamilton

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