Physics 422 -- Fall 2007

Homework #3, due Wednesday Sep. 19 at beginning of class

1. [5pts] (a) Find the Hamiltonian (as a function of its appropriate variables) for the problem given in Marion & Thornton problem 7.4. Use polar coordinates r and theta.
(b) Use the Hamiltonian to find the 4 equations of motion.
(c) Use the Hamiltonian to obtain two constants of the motion.
(d) Use those two constants of motion to express r as a function of time (or time as a function of r) in the form of an integral.

2. [5pts] (a) Find the equation of motion for the apparatus in Marion & Thornton problem 7.18 using the Lagrangian method. Use the angle of the massless string with respect to vertical as your coordinate variable. Do not assume a small angle approximation, and ignore the stuff about theta1 and theta2.
(Hint: You need to compute the kinetic and potential energies. One way to do that is to first calculate the Cartesian coordinates (x,y) of the mass as a function of theta, and then take derivatives to get the velocity.)
(b) Find the Hamiltonian and use it to obtain Hamilton's equations of motion.
(c) Find the frequency for small oscillations.

3. [5pts] Consider a pendulum that consists of a mass M hanging from a massless string of length r. The string is being pulled upward at constant velocity through a tiny hole in the ceiling, so the length of the pendulum is given by r = r0 - alpha*t. Let theta be the angle of the string with respect to vertical. Assume that the motion is in a plane, but do not make small angle approximations.
(a) Find the Lagrange equation of motion.
(b) Find the Hamilton equations of motion.

4. [5pts] Marion & Thornton problem 7.22:
(a) Find the Lagrangian and the Lagrange Equations of motion. (You will first have to find a potential V(x) that corresponds to the given force law.)
(b) Find the Hamiltonian and the Hamiltonian Equations of motion.
(c) Is the Hamiltonian equal to T+V?
(d) Is the Hamiltonian a constant of the motion?