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 on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket.

Definition

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 may be identified with vector fields and every differentiable function determines a unique vector field, called the Hamiltonian vector field with the Hamiltonian,

.

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

-dimensional symplectic manifold. Then locally, one may choose canonical coordinates, in which the symplectic form is expressed as:

denotes the exterior derivative denotes the exterior product takes the form:

square matrix

and

.

-dimensional symplectic vector space with (global) canonical coordinates.

Properties

• is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.

• is an integral curve if and only if it is a solution of Hamilton's equations:

• . This property corresponds to the conservation of energy in Hamiltonian mechanics.

• have a zero Poisson bracket. 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-symmetric, the Poisson bracket, defined by the formula

denotes the Lie derivative. Moreover, one can check that the following identity holds:,

. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:,

, endowed with the Poisson bracket, has the structure of a Lie algebra is a Lie algebra homomorphism, whose kernel is connected).

Remarks

Notes

Works cited

• See section 3.2.

External links

• Hamiltonian vector field on nLab