Assignment 4

Problem 1:

From the Hamiltonian of a free, non-relativistic, classical mass-point, H = p2/(2m0),
(a)  find the equation of motion dp/dt and dq/dt.
(b)  Compute the total differential for an arbitrary function F(p, q, t) and express in terms of Poisson brackets.

Recall that the Poisson bracket
{a, b}PB = ∑k(∂a/∂qk ∂b/∂pk - ∂a/∂pk ∂b/∂qk).

Problem 2:

A point particle with mass m is restricted to move on the inside surface of a horizontal ring.  The radius of the ring increases steadily with time as R = R0 + R't.
At t = 0 the speed of the particle is v0.
(a)  Are there any constants of motion?
(b)  Let E(t1) be the energy of the particle at the time the radius of the ring is 2R0 and E0 its energy at t = 0. 
Find the ration E(t1)/E0.

Problem 3:

A bead is constrained to move without friction on a helix whose equation in cylindrical polar coordinates is ρ  = b, z = aΦ under the influence of the potential V = ½k(ρ2 + z2).  
(a)  Use the Lagrange multiplier method and find the appropriate Lagrangian including terms expressing the constraints. 
(b)  Apply the Euler-Lagrange equations to obtain the equations of motion. 
(c)  Next, repeat parts (a) and (b) without using the Lagrange multiplier method.  Instead, build the constraints into the general coordinate(s).

Problem 4:

An very long rod is being rotated in a vertical plane at a constant angular velocity ω about a fixed horizontal axis (the z-axis) passing through the origin.  The angular velocity is maintained at the value ω for all times by an external agent.  At t = 0 the rod passes through zero-inclination, (θ = 0 at t = 0) where θ is the angle the rod makes with the x-axis.  There is a mass m on the rod.  The mass' coordinates and velocity components at t = 0 are r(0) = g/(2ω2), θ(0) = 0,  dr/dt|0 = 0,   dθ/dt|0 = ω,
where g is the acceleration due to gravity.  The mass m is free to slide along the rod.  Neglect friction. 
Hint: Recall that in plane polar coordinates the unit vectors r/rand θ/θ are not constant.

(a)  Find an expression for r(t), the radial coordinate of the mass, which holds as long as the mass remains on the rod.
(b)  Show that r(t) > r(0) for small t (t  > 0).
(c)  There is a component of the mass' weight acting down the inclined rod, but no force component acting up the rod.  With this in mind, explain why the mass begins moving farther out along the rod instead of down the rod.

Problem 5:

(a)  If the Hamiltonian of a system is given by H = (1/b)pb, with with b = constant, find the corresponding Lagrangian.
(b)  If a system has a Lagrangian L = ½G(q,t)(dq/dt)2 + F(q,t)(dq/dt) - U(q,t),
show that the Hamiltonian can be written
H = (p - F(q,t))2/(2G(q,t)) + U(q,t),
where p = G(dq/dt) + F.