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 H:M\to\mathbb{R} on a symplectic manifold M defines a unique vector field, XH, called a Hamiltonian vector field. It is defined such that for every vector field Y on M the identity

dH(Y) = ω(XH,Y)

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

\omega=\sum_n dq^i \wedge dp_i

and thus the Hamiltonian vector field takes the form

X_H = \left( \frac{\partial H}{\partial p_i}, - \frac{\partial H}{\partial q^i} \right) = \Omega \cdot dH

where Ω is the canonical symplectic matrix

\Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}.

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

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

and

\dot{p}_i = - \frac {\partial H}{\partial q^i}.

Note that the energy is a constant along the integral curve, that is, H(γ(t)) 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

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

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

References

See also: Symplectic vector field, Canonical coordinates, Cotangent bundle, Hamilton-Jacobi equations, Lie algebra, Lie derivative, Mathematics, One-form, One-to-one correspondence, Physics