# Pendulum on a spring accelerating upward

1. Dec 12, 2011

### MiloMinder

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

L = T - U
H = T + U

3. 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?)