Problem 1:

Two spheres are of the same radius R and mass M, but one is solid and the other is a hollow shell (of negligible thickness).  Both spheres roll (without sliding) down a ramp of incline q.
(a)  Which sphere will have the greater acceleration down the ramp?
(b)  Determine the Lagrangian for the motion of the sphere and derive the equation of motion for both cases.

Problem 2:

Obtain Lagrange's equations of motion for a spherical pendulum (a mass point suspended by a rigid, weightless rod).

Problem 3:

The Lagrangian for a simple spring is given by

Find the Hamiltonian and the equations of motion using the Hamiltonian formulation.  Identify any conserved quantities.

Problem 4:

A particle of mass m moves in one dimension under the influence of a force

F(x,t) = -k x exp(-t/t)

where k and t are positive constants. (Note carefully the dependence on x to the first power).
(a) Compute the Lagrangian function.
(b) Use Lagrange’s equation to determine the equation of motion explicitly.
(c) Compute the Hamiltonian function in terms of the generalized coordinate and generalized momentum. (Show clearly how you get this.)
(d) Determine Hamilton’s equations of motion explicitly for this particular problem (not just general formulae).
(e) Does the Hamiltonian equal the total energy?
(f) Is the total energy of the mass conserved?
(g) What is it about the force F which supports your answer to part f?

Problem 5:

A point mass m slides without friction along a wire bent into a vertical circle of radius a.  The wire rotates with constant angular velocity W about the vertical diameter, and the apparatus is placed in a uniform gravitational field g parallel to the axis of rotation.
(a)  Construct the Lagrangian for the point mass using the angle q (angular displacement measured from the downward vertical) as its generalized coordinate and find its equation of motion.
(b)  What are the equilibrium values for q?  For each equilibrium value, what is the condition for the equilibrium to exist?
(c)  For each equilibrium value, is the equilibrium stable or unstable against small displacements along the wire?  What is the oscillation frequency or growth rate of such perturbations?

Problem 6:

A small object with mass m moves on a smooth, friction-free horizontal surface.  It is attached to a peg at the origin by an ideal massless spring with spring constant k and equilibrium length r₀.  At time t = 0, the mass is set in motion in an arbitrary direction from point (r,q).

(a)  Find the Lagrangian L for the system, then
(b)  calculate the generalized momenta p_j .
(c)  Construct the Hamiltonian function, H(p_j, q_j, t);
(d)  then work out the equations of motion dp_j/dt and dq_j/dt. 
(e)  Are any of the variables cyclic, thereby giving especially simple equations of motion?  If so, integrate the equation(s) and interpret your results physically.
(f)  Consider the special case that r = constant.  Deduce the condition(s) that allow this case and discuss how this occurs physically.

Problem 7:

Consider a hoop of mass m and radius r rolling without slipping down an incline.
(a)  Determine the Lagrangian L(x, dx/dt) of this one-degree-of-freedom system.  Derive from it the Lagrange equation and its solution for initial condition x₀ = 0, dx/dt|₀ = 0.
(b)  Determine the alternative Lagrangian L(x, dx/dt, q, dq/dt)  and the holonomic constraint f(x, q) = 0 that must accompany it.  Derive the associated three equations of motion for the two unknown dynamical variables x, q and the undetermined Lagrange multiplier l.  Solve these equations for the same initial conditions as in (a) and determine the static frictional force of constraint between the hoop and the incline.