Hamiltonian dynamics is introduced as an advanced topic at the undergrad level or an introductory topic at the graduate level.

In the example given the Hamiltonian is the sum of a kinetic energy, \(p/2m\), and a potential energy \(V(q)\), but there are many systems ni which the Hamiltonian does not have this form.

Conservative One-degree-of-freedom Hamiltonian Systems

  1. Show that if \(H=H(q,p)\) and that the pair \((q,p)\) obeys Hamilton’s equations of motion, then

\begin{align} \frac{d H}{dt}=0 \end{align}

i.e., that Hamiltonian is conserved, which is why we call such systems conservative. For Newtonian systems, the conserved value of \(H\) is equal ot the total energy \(E\).

Example: The linear oscillator

For the equations of motion to be linear the potential \(V(q)\) must be at most quadratic. Such conditions are satisfied by the

Example: The linear repulsive force

Example: The cubic potential