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