Pendulum on a spring accelerating upward

[MiloMinder]

Homework Statement

A pendulum on Earth consists of a mass m suspended on a massless spring with equilibrium length d and spring constant k. The pendulum point of support is moving up with constant acceleration a.

What is the Hamiltonian of the system? Derive the Hamiltonian equations of motion for the system. Discuss the relationship between the Hamiltonian and the total energy of the system (is energy conserved?). What are the frequencies of small (first order in the variables) oscillations for the system?

Homework Equations

L = T - U
H = T + U

The Attempt at a Solution

I'm using r, θ, and z as my coordinates.
T = 1/2 m [ (rθ'cosθ + r'sinθ)^2 + (-rθ'sinθ + r'cosθ + z')
U = mg (z + rcosθ)

L = T - U

The first part of the problem asks for the Lagrangian equations of motion. I found those by applying Euler's equation to L and solving the differential equations.

Now what is the correct method for finding H? Is it as simple as T + U? Or do I need to derive the momenta for the system (and how would I do that?)

 

[member 553083]

? And if I do find the momenta, how would I use them in my Hamiltonian?For the second part of the problem, I'm guessing that energy is conserved since the equations of motion are derived from the Lagrangian which takes into account the total energy. The third part of the problem is asking for the frequencies of small (first order in the variables) oscillations. I'm not sure how to solve this. Any help would be appreciated. Thanks!