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 \(\rho(\{q_i\},\{ p_i\},t) d^n p d^n q\).

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