11.5 Hamiltonian systems
Definition 11.5.1. A Hamiltonian system with \(n\) degrees of freedom and Hamiltonian \(H(q, p)\), a \(C^2\) function of positions \(q \in \mathbb R^n\) and momenta \(p \in \mathbb R^n\), is \[ q' = \frac {\partial H}{\partial p}, \qquad p' = -\frac {\partial H}{\partial q} . \]
- (i)
- \(H\) is constant along every solution.
- (ii)
- The flow preserves volume in the \(2n\)-dimensional phase space.
- (iii)
- No equilibrium is asymptotically stable.
- (iv)
- An equilibrium at a strict local minimum of \(H\) is stable.
Proof. (i) \(\dot H = H_q\cdot q' + H_p\cdot p' = H_q\cdot H_p - H_p\cdot H_q = 0\). (ii) The divergence of the vector field is \(\sum _i\bigl (\partial ^2H/\partial q_i\partial p_i - \partial ^2H/\partial p_i\partial q_i\bigr ) = 0\), and by Liouville’s formula applied to the variational equation (Section 4.2) the Jacobian determinant of the flow is \(1\). (iii) If an equilibrium were asymptotically stable, a small ball around it would be mapped by \(\phi _t\) into an ever smaller neighbourhood, so its volume would shrink, contradicting (ii). (iv) \(V = H - H(x^*)\) is positive definite near \(x^*\) and \(\dot V = 0\), so Theorem 11.1.2 gives stability. □
The undamped pendulum, \(H = \frac 12p^2 + 1 - \cos q\), is Hamiltonian: the hanging equilibria are minima of \(H\), stable but not asymptotically stable (they are centres surrounded by closed orbits), and the upright ones are saddles of \(H\). The Lotka–Volterra system of Theorem 10.5.1 becomes Hamiltonian in the variables \(\log x\) and \(\log y\). Volume preservation has a numerical consequence: a numerical method that does not respect it, such as explicit Euler, slowly pumps energy into or out of the system, while symplectic methods, which preserve the Hamiltonian structure, keep the energy error bounded for very long times (Software 11.6.3).
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