Exercises

1. A particle of mass $m$ is placed at the top of a smooth vertical hoop of radius $a$. Calculate the reaction of the hoop on the particle as it slides down the hoop by means of the method of Lagrange multipliers. Find the height at which the particle falls off the hoop.
2. A uniform disk of mass $m$ and radius $a$ has a light string wrapped around its circumference with one end of the string attached to a fixed support. The disk is allowed to fall under gravity, unwinding the string as it falls. Solve the problem using the method of Lagrange multipliers. What is the tension in the string?
3. Consider two particles of masses $m_1$ and $m_2$. Let $m_1$ be constrained to move on a circle of radius $a$ in the $z=0$ plane, centered at $x=y=0$. (Here, $z$ measures vertical height). Let $m_2$ be constrained to move on a circle of radius $a$ in the $z=c$ plane, centered on $x=y=0$. A light spring of spring constant $k$ and unstretched length $c$ is attached between the particles. Find the Lagrangian of the system. Solve the problem using Lagrange multipliers and give a physical interpretation for each multiplier.

4. Find the Hamiltonian of a particle of mass $m$ constrained to move under gravity on the inside of a sphere of radius $a$. Use the standard spherical polar coordinates $\theta$ and $\phi$ as your generalized coordinates, where the axis of the coordinates points vertically downward. Find Hamilton's equations of motion for the system.
5. A particle of mass $m$ is subject to a central attractive force given by
   
   $${\bf f} = - \frac{k\,{\rm e}^{-\beta\,t}}{r^2}\,{\bf e}_r,$$
   
   
   where $k$ and $\beta$ are positive constants. Find the Hamiltonian of the particle. Compare the Hamiltonian to the total energy of the particle. Is the energy of the particle conserved?