- #1
Clau
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Homework Statement
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.
Homework 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]
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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