Hamiltonian Vector Field - Examples

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

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

\Chi_H=\left( \frac{\partial H}{\partial p_i}, - \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,

where Ω is a 2n by 2n square matrix

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

and

\mathrm{d}H=\begin{bmatrix} \frac{\partial H}{\partial q^i} \\ \frac{\partial H}{\partial p_i} \end{bmatrix}.

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

  • If H = pi then
  • if H = qi then
  • if then
  • if then

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