Homework

1. A massless spring of unstretched length $\ell$ and spring constant $k$ connects two masses $m_1$ and $m_2$. The system floats in space with no external forces. Other than the spring connection, the two masses are free to move in any direction (6 coordinates). Write the Lagrangian for the system. Find as many conserved momenta as possible by choosing the best coordinates you can to describe the system. Write Hamilton's equations of motion for this system.

2. A spherical pendulum consists of a bob of mass $m$ attached to a massless extensionless rod of length $\ell$. The other end of the rod is attached to a fixed frictionless pivot point. Find the equation of motion using the Hamiltonian using the conserved momentum and to derive an effective potential. Sketch the potential as a function of $\theta$ for several values of the momentum $p_\phi=0$. Under what conditions will the motion be at a constant $\theta$?

3. Consider a beam of electrons moving in the $z$ direction with constant intensity inside a circle of radius $a$. The beam is made to have only a small but important spread in transverse momentum. Assume that the momentum is also of uniform density over a circle of radius $b$ in momentum space. Before the beam can spread out, a focusing element focuses the beam down to a circle of radius $a_{focus}$ in the $xy$ plane. Assume that this a measured at the point where the radius is a minimum. What will the distribution of transverse momenta be at the focus? Draw a diagram of the motion of the particles which includes the region before the focusing element, after it, and after the focus.