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.

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

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

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

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

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

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

$${\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}$$

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

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

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

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

and

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

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

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

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

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

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

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

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

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

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

• Hamiltonian vector field on nLab