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 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 between the tangent bundle and the cotangent bundle, with the inverse 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, 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 where denotes the exterior derivative and denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian takes the form where is a square matrix and The matrix is frequently denoted with. Suppose that M = R2n is the 2n-dimensional symplectic vector space with canonical coordinates.
If then
if then
if then
if then
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 . Then a curve is an integral curve of the Hamiltonian vector field if and only if it is a solution of Hamilton's equations:
More generally, if two functions and have a zero Poisson bracket, 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.
The symplectic form is preserved by the Hamiltonian flow. Equivalently, the Lie derivative
Poisson bracket
The notion of a Hamiltonian vector field leads to a skew-symmetricbilinear operation on the differentiable functions on a symplectic manifold M, the Poisson bracket, defined by the formula where denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds: where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence, the Poisson bracket satisfies the Jacobi identity 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.
