level sets of an Hamiltonian

Question

Consider an hamiltonian system

$$\frac{\partial H}{\partial p} = \dot{x}$$ $$\frac{\partial H}{\partial x} = -\dot{p}$$

what is the dynamical significance of the level sets of the Hamiltonian $H(x,p) = h$?

Answer 1

The point of it is that $H(x,p)$ is constant: $$ \dfrac{dH}{dt} = \dfrac{\partial H}{\partial p} \dot{p} + \dfrac{\partial H}{\partial x} \dot{x} = 0$$ Thus the trajectories of the system go along these level curves.

EDIT: For example, consider the harmonic oscillator which has $H(x,p) = x^2 + p^2$. The general solution of the system $$ \dot{x} = 2 p,\ \dot{p} = - 2 x $$ can be written as $x = r \sin(2(t - t_0))$, $p = r \cos(2(t - t_0))$ which moves around the circle $H(x,p) = x^2 + p^2 = r^2$.