Hamiltonian Dynamics
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Hamiltonian equations are
.. math::
\dot q_i &= \frac{\partial H}{\partial p_i} \\
\dot p_i &= -\frac{\partial H}{\partial q_i}.
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 $\rho(\{q_i\}, \{p_i\}, t)$,
.. math::
\frac{d\rho}{dt} = 0.
.. admonition:: Phase Space Density
:class: notes
The probability that the system will be found in a phase space interval $d^n p d^n q$ is given by $\rho(\{q_i\},\{ p_i\},t) d^n p d^n q$.