Review / Part II — Higher-Order & Nonlinear / Ch 14

Monotone & Conserved Quantities

Ch 14 — Part II — Higher-Order & Nonlinear

A conserved quantity (or first integral) \(H(\mathbf{x})\) is a function constant along trajectories: \(\dot H = \nabla H \cdot \mathbf{F} = 0\). Conserved quantities turn systems into level sets and often explain nonlinear centers.

Hamiltonian systems

If the system has form \(\dot x = \partial H/\partial y,\; \dot y = -\partial H/\partial x\), then \(H\) is conserved. Trajectories lie on level curves of \(H\).

Monotone quantities (Lyapunov)

If \(L(\mathbf{x}) \ge 0\) with \(L(\mathbf{x}^*) = 0\) and \(\dot L \le 0\) along orbits, \(\mathbf{x}^*\) is (Lyapunov) stable. Strict inequality gives asymptotic stability.

Undamped oscillator example

For \(\ddot x + \omega^2 x = 0\), energy \(H = \tfrac{1}{2}\dot x^2 + \tfrac{1}{2}\omega^2 x^2\) is conserved; trajectories are ellipses.

Practice Problems

Problem 1easy
Show that \(E(x, v) = \tfrac{1}{2} v^2 + U(x)\) is conserved for \(\ddot x = -U'(x)\).
Hint
Compute \(dE/dt\).
Solution

\(dE/dt = v \dot v + U'(x) \dot x = v(-U'(x)) + U'(x) v = 0\).

Answer: Conserved.

Problem 2medium
Given \(\dot x = y,\; \dot y = -x - x^3\), find a conserved quantity.
Hint
Treat as Hamiltonian: \(H = \tfrac{1}{2} y^2 + \tfrac{1}{2} x^2 + \tfrac{1}{4} x^4\).
Solution

Verify: \(\dot H = y\dot y + (x + x^3)\dot x = y(-x - x^3) + (x+x^3) y = 0\).

Answer: \(H = \tfrac{y^2}{2} + \tfrac{x^2}{2} + \tfrac{x^4}{4}.\)

Problem 3medium
Use the conserved energy of the simple pendulum \(\ddot\theta + \sin\theta = 0\) to decide whether (0,0) is Lyapunov stable.
Hint
\(E = \tfrac{1}{2}\dot\theta^2 + (1 - \cos\theta)\) is conserved, minimum at origin.
Solution

\(E \ge 0\), zero only at origin; \(\dot E = 0\). Origin is Lyapunov stable but not asymptotically stable (orbits don't converge to origin).

Answer: Lyapunov stable (not asymptotic).