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.Template:Sfn

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

Definition

Suppose that Template:Math is a symplectic manifold. Since the symplectic form Template:Math is nondegenerate, it sets up a fiberwise-linear isomorphism

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

between the tangent bundle Template:Math and the cotangent bundle Template:Math, with the inverse

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

Therefore, one-forms on a symplectic manifold Template:Math may be identified with vector fields and every differentiable function Template:Math determines a unique vector field Template:Math, called the Hamiltonian vector field with the Hamiltonian Template:Math, by defining for every vector field Template:Math on Template:Math,

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

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.

Examples

Suppose that Template:Math is a Template:Math-dimensional symplectic manifold. Then locally, one may choose canonical coordinates Template:Math on Template:Math, in which the symplectic form is expressed as:Template:Sfn $\omega =\sum _i\mathrm{d}{q}^i\wedge \mathrm{d}{p}_i,$

where Template:Math denotes the exterior derivative and Template:Math denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian Template:Math takes the form:Template:Sfn ${\mathrm{X}}_H=(\frac{\partial H}{\partial p_i},-\frac{\partial H}{\partial q^i})=\mathrm{\Omega }\mathrm{d}H,$

where Template:Math is a Template:Math square matrix

$$\mathrm{\Omega }=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right],$$

and

$$\mathrm{d}H=\left[\begin{array}{c}\frac{\partial H}{\partial q^i}\\ \frac{\partial H}{\partial p_i}\end{array}\right].$$

The matrix Template:Math is frequently denoted with Template:Math.

Suppose that M = R²ⁿ is the 2n-dimensional symplectic vector space with (global) canonical coordinates.

• If $H=p_i$ then $X_H=\partial /\partial q^i;$
• if $H=q_i$ then $X_H=-\partial /\partial p^i;$
• if $H=\frac{1}{2}\sum (p_i{)}^2$ then $X_H=\sum p_i\partial /\partial q^i;$
• if $H=\frac{1}{2}\sum a_{ij}{q}^i{q}^j,a_{ij}=a_{ji}$ then $X_H=-\sum a_{ij}{q}_i\partial /\partial p^j.$

Properties

• The assignment Template:Math is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that Template:Math are canonical coordinates on Template:Math (see above). Then a curve Template:Math is an integral curve of the Hamiltonian vector field Template:Math if and only if it is a solution of Hamilton's equations:Template:Sfn $$\begin{array}{cc}{\dot{q}}^i& =\frac{\partial H}{\partial p_i}\\ {\dot{p}}_i& =-\frac{\partial H}{\partial q^i}.\end{array}$$
• The Hamiltonian Template:Math is constant along the integral curves, because $⟨dH,\dot{\gamma }⟩=\omega (X_H(\gamma ),X_H(\gamma ))=0$. That is, Template:Math is actually independent of Template:Math. This property corresponds to the conservation of energy in Hamiltonian mechanics.
• More generally, if two functions Template:Math and Template:Math have a zero Poisson bracket (cf. below), then Template:Math is constant along the integral curves of Template:Math, and similarly, Template:Math is constant along the integral curves of Template:Math. This fact is the abstract mathematical principle behind Noether's theorem.
• The symplectic form Template:Mvar is preserved by the Hamiltonian flow. Equivalently, the Lie derivative ${ℒ}_{X_H}\omega =0.$

Poisson bracket

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

$$\{f,g\}=\omega (X_g,X_f)=dg(X_f)={ℒ}_{X_f}g$$

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

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

which means that the vector space of differentiable functions on Template:Math, endowed with the Poisson bracket, has the structure of a Lie algebra over Template:Math, and the assignment Template:Math is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if Template:Math is connected).

Works cited

• Template:Cite bookSee section 3.2.
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External links

• Hamiltonian vector field on nLab