References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 2.5; Exercises 2.5.2 – 2.5.3.
Here are a couple of examples of equations of motion using the Hamiltonian formalism. First, we look at the simple harmonic oscillator, in which we have a mass sliding on a frictionless horizontal surface. The mass is connected to a spring with constant , with the other end of the spring connected to a fixed support.
The Hamiltonian is given by
where the velocities are expressed in terms of the positions and momenta . In this case, we have, using the coordinate as the displacement from equilibrium
We can now apply Hamilton’s canonical equations:
We thus get a pair of first order ODEs which can be solved in the usual way, given and . The second order ODE that we got by using the Lagrangian method can be obtained by differentiating the first equation and plugging it into the second:
From 7 we see that, since in the absence of external force, the total energy is a constant,
This can be written as the equation of an ellipse:
We can use the Hamiltonian formalism to get the equations of motion of the coupled harmonic oscillator. From our Lagrangian treatment, we had
Converting to coordinates and momenta, we have
Applying the canonical equations gives
Again, by taking the derivative of the first line and substituting into the last two lines, we get back the previous equations of motion: