6.9: A Simple Example
( \newcommand{\kernel}{\mathrm{null}\,}\)
For a particle moving in a potential in one dimension,
Hence
Therefore
(Of course, this is just the total energy, as we expect.)
The Hamiltonian equations of motion are
So, as we’ve said, the second order Lagrangian equation of motion is replaced by two first order Hamiltonian equations. Of course, they amount to the same thing (as they must!): differentiating the first equation and substituting in the second gives immediately
