From the Hamiltonian of a free, non-relativistic, classical mass-point, H =
**p**^{2}/(2m_{0}),

(a) find the equation of motion d**p**/dt and d**q**/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/∂q_{k} ∂b/∂p_{k} - ∂a/∂p_{k}
∂b/∂q_{k}).

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
= R_{0} + R't.

At t = 0 the speed of the particle is v_{0}.

(a) Are there any constants of motion?

(b) Let E(t_{1}) be the energy of the particle at the time the
radius of the ring is 2R_{0} and E_{0} its energy at t = 0.

Find the ration E(t_{1})/E_{0}.

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(ρ^{2} + z^{2}).

(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).

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/(2ω^{2}), 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.

(a) If the Hamiltonian of a system is given by H = (1/b)p^{b},
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.