3 Properties of Hamiltonian Systems

If we interprete the functional (the Hamiltonian)

\[ H(p,q) \]

as the energy and define suitable trajectories \((p,q)\) as those that preserve the energy by requiring

\(H(p(t),q(t)) = const.\)

we can infer that

  • \((p,q)\) describe path along the isolines of \(H\),
  • the velocities \((\dot p, \dot q)\) are orthogonal to the gradient \(\nabla H\),
  • or that
\[\begin{equation*} \begin{split} \dot p &= - \frac{\partial H}{\partial q} \\ \dot q &= \phantom{-} \frac{\partial H}{\partial p} \end{split} \end{equation*}\]

which we wrote in compact form as

\[ \begin{bmatrix} \dot p \\ \dot q \end{bmatrix} = J^{-1}\nabla H \]

with the structure matrix

\[ J^{-1} = \begin{bmatrix} 0 & -I \\ I & 0 \end{bmatrix}. \]

Note that \(J^T = -J = J^{-1}\).