Symplectic vector field

In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. The integral curves of the symplectic vector field are solutions to the Hamilton-Jacobi equations of motion. The vector field, taken together with the symplectic manifold and the symplectic form on the manifold, comprise a Hamiltonian system.

Definition

Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorphism between the tangent bundle and the cotangent bundle, thus establishing a one-to-one correspondence between tangent vectors and one-forms. As a special case, every differentiable function <math>H:M\to\mathbb{R}<math> on a symplectic manifold M defines a unique vector field, X_H, called a Hamiltonian vector field. It is defined such that for every vector field Y on M the identity

dH(Y) = ω(X_H,Y)

holds. In canonical coordinates <math>(q^1,\ldots ,q^n,p_1,\ldots,p_n)<math>, the symplectic form can be written as

<math>\omega=\sum_n dq^i \wedge dp_i<math>

and thus the Hamiltonian vector field takes the form

<math>X_H = \left( \frac{\partial H}{\partial p_i},

- \frac{\partial H}{\partial q^i} \right) = \Omega \cdot dH<math>

where Ω is the canonical symplectic matrix

<math>\Omega =

\begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}<math>.

A curve <math>\gamma(t)=(q(t),p(t))<math> is thus an integral curve of the vector field if and only if it is a solution of the Hamilton-Jacobi equations:

<math>\dot{q}^i = \frac {\partial H}{\partial p_i}<math>

and

<math>\dot{p}_i = – \frac {\partial H}{\partial q^i}<math>.

Note that the energy is a constant along the integral curve, that is, <math>H(\gamma(t))<math> is a constant independent of t.

Poisson bracket

The Hamiltonian vector fields give differentiable functions on M the structure of a Lie algebra with bracket the Poisson bracket

<math>\{f,g\} = \omega(X_f,X_g)= X_g(f) = \mathcal{L}_{X_g} f<math>

where <math>\mathcal{L}_X<math> is the Lie derivative along X. Note that some authors use sign conventions that differ from the above.

References

• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0–198–50451–9.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0–8053–0102-X See section 3.2.