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 ''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 is a symplectic manifold. Since the symplectic form is nondegenerate, it sets up a ''fiberwise-linear'' isomorphism

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

between the tangent bundle and the cotangent bundle , with the inverse

: $\Omega:T^*M\to TM, \quad \Omega=\omega^.$

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

:$\mathrmH(Y) = \omega(X_H,Y).$

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 is a -dimensional symplectic manifold. Then locally, one may choose canonical coordinates on , in which the symplectic form is expressed as: $\omega=\sum_i \mathrmq^i \wedge \mathrmp_i,$

where denotes the exterior derivative and denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian takes the form: $\Chi_H=\left( \frac, - \frac \right) = \Omega\,\mathrmH,$

where is a square matrix

:$\Omega = \begin 0 & I_n \\ -I_n & 0 \\ \end,$

and

:$\mathrmH=\begin \frac \\ \frac \end.$

The matrix is frequently denoted with .

Suppose that ''M'' = R^2''n'' is the 2''n''-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 $H=1/2\sum (p_i)^2$ then $X_H=\sum p_i\partial/\partial q^i;$
* if $H=1/2\sum a_ q^i q^j, a_=a_$ then $X_H=-\sum a_ q_i\partial/\partial p^j.$

Properties

* The assignment is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
* Suppose that are canonical coordinates on (see above). Then a curve is an integral curve of the Hamiltonian vector field if and only if it is a solution of Hamilton's equations: $\dot^i = \frac$
:$\dot_i = - \frac .$

* The Hamiltonian is constant along the integral curves, because $\langle dH, \dot\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0$. That is, is actually independent of . This property corresponds to the conservation of energy in Hamiltonian mechanics.
* More generally, if two functions and have a zero Poisson bracket (cf. below), then is constant along the integral curves of , and similarly, is constant along the integral curves of . This fact is the abstract mathematical principle behind Noether's theorem.See for a very concise statement and proof of Noether's theorem.
* The symplectic form is preserved by the Hamiltonian flow. Equivalently, the Lie derivative $\mathcal_ \omega= 0.$

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

:$\ = \omega(X_g, X_f)= dg(X_f) = \mathcal_ g$

where $\mathcal_X$ denotes the Lie derivative along a vector field ''X''. Moreover, one can check that the following identity holds: 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: $\+\+\=0,$

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

Remarks

Notes

Works cited

*''See section 3.2''.
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Hamiltonian mechanics
Symplectic geometry
William Rowan Hamilton