Hamiltonian Dynamics¶

Hamiltonian equations are

$\begin{split}\dot q_i &= \frac{\partial H}{\partial p_i} \\ \dot p_i &= -\frac{\partial H}{\partial q_i}.\end{split}$

Some constant of motion can be read out from the equations by recogonizing the fact that the time derivative of a constant of motion, $$q_i$$ or $$p_i$$, is zero. For example, if the Hamiltonian doesn’t explicitly depend on $$p_k$$, we have $$\frac{\partial H}{\partial p_k} = 0 = \dot q_k$$, which means that $$q_k$$ is a constant of motion.

The evolution of the system in phase space obeys the Liouville’s theorem, which describes the motion of phase space density $$\rho(\{q_i\}, \{p_i\}, t)$$,

$\frac{d\rho}{dt} = 0.$

Phase Space Density

The probability that the system will be found in a phase space interval $$d^n p d^n q$$ is given by $$\rho(\{q_i\},\{ p_i\},t) d^n p d^n q$$.