Hamiltonian vector field

Template:Short description 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 $(M,\omega )$ is a symplectic manifold. Since the symplectic form $\omega$ is nondegenerate, it sets up a fiberwise-linear isomorphism

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

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

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

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

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

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 $M$ is a $2n$-dimensional symplectic manifold. Then locally, one may choose canonical coordinates $(q^1,\cdots ,q^n,p_1,\cdots ,p_n)$ on $M$, in which the symplectic form is expressed as:Template:Sfn $\omega =\sum _i\mathrm{d}{q}^i\wedge \mathrm{d}{p}_i,$

where $\mathrm{d}$ denotes the exterior derivative and $\wedge$ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian $H$ 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 $\mathrm{\Omega }$ is a $2n\times 2n$ 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 $\mathrm{\Omega }$ is frequently denoted with $\mathbf{𝐉}$.

Suppose that $M={\mathbb{ℝ}}^{2n}$ 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 $f\mapsto X_f$ is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that $(q^1,\cdots ,q^n,p_1,\cdots ,p_n)$ are canonical coordinates on $M$ (see above). Then a curve $\gamma (t)=(q(t),p(t))$ is an integral curve of the Hamiltonian vector field $X_H$ if and only if it is a solution of Hamilton's equations:Template:Sfn $$\begin{array}{rl}{\dot{q}}^i& =\frac{\partial H}{\partial p_i}\\ {\dot{p}}_i& =-\frac{\partial H}{\partial q^i}.\end{array}$$
• The Hamiltonian $H$ is constant along the integral curves, because $⟨dH,\dot{\gamma }⟩=\omega (X_H(\gamma ),X_H(\gamma ))=0$. That is, $H(\gamma (t))$ is actually independent of $t$. This property corresponds to the conservation of energy in Hamiltonian mechanics.
• More generally, if two functions $F$ and $H$ have a zero Poisson bracket (cf. below), then $F$ is constant along the integral curves of $H$, and similarly, $H$ is constant along the integral curves of $F$. This fact is the abstract mathematical principle behind Noether's theorem.
• The symplectic form $\omega$ 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 $M$, endowed with the Poisson bracket, has the structure of a Lie algebra over $\mathbb{ℝ}$, and the assignment $f\mapsto X_f$ is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if $M$ is connected).

Works cited

• See section 3.2.

External links

• Hamiltonian vector field on nLab