 # Hamiltonian vector field

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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.

## Contents

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

ω : T M T M ,

between the tangent bundle TM and the cotangent bundle T*M, with the inverse

Ω : T M T M , Ω = ω 1 .

Therefore, one-forms on a symplectic manifold M may be identified with vector fields and every differentiable function H: MR determines a unique vector field XH, called the Hamiltonian vector field with the Hamiltonian H, by requiring that for every vector field Y on M, the identity

d H ( Y ) = ω ( X H , Y )

must hold.

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 (q1, ..., qn, p1, ..., pn) on M, in which the symplectic form is expressed as

ω = i d q i d p i ,

where d denotes the exterior derivative and ∧ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H takes the form

X H = ( H p i , H q i ) = Ω d H ,

where Ω is a 2n by 2n square matrix

Ω = [ 0 I n I n 0 ] ,

and

d H = [ H q i H p i ] .

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

• If H = pi then X H = / q i ;
• if H = qi then X H = / p i ;
• if H = 1 / 2 ( p i ) 2 then X H = p i / q i ;
• if H = 1 / 2 a i j q i q j , a i j = a j i then X H = a i j q i / p j .
• ## Properties

• The assignment fXf is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that (q1, ..., qn, p1, ..., pn) are canonical coordinates on M (see above). Then a curve γ(t)=(q(t),p(t)) is an integral curve of the Hamiltonian vector field XH if and only if it is a solution of the Hamilton's equations:
• q ˙ i = H p i p ˙ i = H q i .
• The Hamiltonian H is constant along the integral curves, because d H , γ ˙ = ω ( X H ( γ ) , X H ( γ ) ) = 0 . That is, H(γ(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 ω is preserved by the Hamiltonian flow. Equivalently, the Lie derivative L X H ω = 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 } = ω ( X g , X f ) = d g ( 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:

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

{ { 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 R, and the assignment fXf is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M is connected).

## References

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