Problem 1:

A  block with mass m is hanging on a spring with spring constant k. 
The spring is held on the other side by a hand. 

(a)  If the hand moves as  y_h(t) = Asin(ωt), use the Lagrangian formalism to find the equation of motion of the block?  Assume that the spring is ideal and there is no damping.  Assume that ω is not the natural frequency of the spring."
(b)  What "initial conditions" at t = 0 are needed so that the block oscillates with a single frequency ω?  Under those conditions, what is the oscillation amplitude of the block?  What is the oscillation phase of the block relative to the hand?

Problem 2:

Consider the N equally spaced, identical balls connected by springs.  All balls have mass m, all springs have spring constant β, and adjacent balls are separated by distance a at equilibrium.

(a)  Write down the equation of motion for each ball.  You can assume periodic boundary conditions.

(b)  Longitudinal wave can propagate along the springs and balls. 
Prove that the dispersion relation between the wave angular frequency ω and wave vector k is given by
ω_n = 2(β/m)^½|sin(k_n*a/2)|.
Here ω₀² = 4β/m.

Problem 3:

A particle of mass m is moving on the surface of a cone placed vertically with the vertex downwards in a uniform gravitational field.  The vertical angle of the cone is 2α .
(a)  Obtain the Lagrange equations of motion.

(b)  What quantities are conserved?  What is the energy of the system?

(c)  Determine the number of physical turning points in the orbit.  That is, does an infalling particle plunge, bounce to infinity, or oscillate back and forth in distance?
Show your work.

Problem 4:

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,φ).  Assume the spring can rotate about the origin, but not bend.

(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.