(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

According to Liouville's theorem, the motion of phase-space points defined by Hamilton's equations conserves phase-space volume. The Hamiltonian for a single particle in one dimension, subjected to a constant force F, is

[tex]H(x,p_{x}) = \frac{p_{x}^2}{2.m} - F.x[/tex]

Consider the phase space rectangle of initial points defined by

0 < x < A and 0 < p < B

Let the points in the rectangle move according to Hamilton's equations for a time t and sketch how the rectangle changes with time in the [tex]p_{x}[/tex]-x plane.

2. Relevant equations

[tex] \frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^d\left(\frac{\partial\rho}{\partial q^i}\dot{q}^i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0. [/tex]

3. The attempt at a solution

Substituting the Hamiltonian from the problem inside the Liouville's equation I can see that the density of particles of this volume is constant.

But, I don't know how to show the movement of this rectangle with time.

I guess that there is no difference...

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# Homework Help: Liouville's theorem

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