Essence: Can the change in phase space volume, due to the flow, in general be periodic (for Hamiltonian systems)?

Specific case: Consider

$$(1+e\cos t)\frac{d^2\theta}{dt^2}-2e\sin t\frac{d\theta}{dt}-\frac{w^2}{2}\sin 2(t-\theta)=0,$$ where $0<e<1$, $w>0$. Its Hamiltonian is

$$H=\frac{1}{2}(1+e\cos t)^2\dot{\theta}^2-\frac{1}{4}w^2(1+e\cos t)\cos 2(t-\theta)$$

In dynamical systems form:

$$\dot{\theta}=\frac{d\theta}{dt}=\eta\equiv F_1(\theta,\eta)$$ $$\dot{\eta}=\frac{d\eta}{dt}=\frac{2 e \sin t \cdot \eta-w^2 \sin 2(t-\theta)}{1+e \cos t}\equiv F_2(\theta,\eta),$$

and $F=F(\theta,\eta)=(F_1,F_2)$.

Thence, via

$$\frac{d}{dt}\ln V(t)=\nabla \cdot F\equiv\frac{\partial F_1}{\partial \theta}+\frac{\partial F_2}{\partial \eta}$$

I get

$$V(t)=V_0\frac{1}{(1+e\cos t)^2}.$$

On the other hand, I'd expect a ball of initial conditions to spread out in the (available parts of) phase space due to topological mixing. Is this in agreement/contradictory with my periodic volume?


You are possibly mixing Lagrangian formulation and Hamiltonian formulation? In $(\theta,\dot{\theta})$ coordinates your system need not preserve phase space volume. In $(\theta,p)$ coordinates it does. What is your $p$ ? If the kinetic energy is $$ K_t(\dot{\theta}) = \frac12 (1+e \cos(t))^2 \dot{\theta}^2$$ Then $ p = \frac{\partial K}{\partial \dot{\theta}} =\dot{\theta}(1+e \cos(t))^2$ and the Hamiltonian reads: $H_t(p,\theta) = \frac{p^2}{2(1+e \cos(t))^2}+ ...$

Phase space volume invariance leads to $$dp_t\wedge d\theta_t = dp_0 \wedge d\theta_0 \Leftrightarrow d\dot{\theta}_t \wedge d\theta_t (1+e \cos(t))^2 = d\dot{\theta}_0 \wedge d\theta_0 (1+e)^2$$ writing $V_t$ for the latter you get the mentioned identity (a part from the factor $(1+e)^2$). I can't think of any interesting conclusion from the fact that the latter is oscillating (a part from the fact that the kinetic energy must depend explicitly upon time).

Note that chaos/mixing is not related to phase space conservation. A little 'ball' may be stretched in some direction, compressed in another leaving the volume invariant.


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