Homework 4
Last date: Dec 28
Questions
1. Consider a mass m moving in two dimensions with potential energy
U (x, y) = 12 kr2 , where r2 = x2 + y 2 . Write down the Lagrangian, using
coordinates x and y, and find the two Lagrange equations of motion.
Describe their solutions.
2. (a) Write down the Lagrangian L(x1 , x2 , ẋ1 , ẋ2 ) for two particles of
equal masses m1 = m2 = m, confined to the x-axis and connected by
a spring with potential energy U = 21 kx2 . Here x = (x1 − x2 − l),
where l is the spring’s unstretched length. (b) Rewrite L in terms of
the new variables X = 21 (x1 + x2 ) (the center of mass position) and x
(the extension). Write down the two Lagrange equations for X and x.
(c) Solve for X(t) and x(t) and describe the motion.
3. Write down the Lagrangian for a cylinder (mass m, radius R, and mo-
ment of inertia I) that rolls without slipping straight down an inclined
plane at an angle α from the horizontal. Use the cylinder’s distance x
down the plane as the generalized coordinate. Write down the Lagrange
equation and solve it for the cylinder’s acceleration ẍ.
4. Consider a bead of mass m sliding without friction on a wire bent in the
shape of a parabola. The wire is spun with constant angular velocity
ω about its vertical axis. Use cylindrical polar coordinates and let the
equation of the parabola be z = kp2 . Write down the Lagrangian in
terms of p as the generalized coordinate. Find the equation of motion of
the bead and determine whether there are equilibrium positions where
the bead can remain fixed. Discuss the stability of these equilibrium
positions.
1
Figure 1:
5. Consider a particle of mass m and charge q moving in a uniform mag-
netic field B in the z-direction. (a) Prove that B = ∇ × A, where
A = 12 B × r. Show equivalently that in cylindrical polar coordinates,
A = 21 Bpϕ̂. (b) Write the Lagrangian in cylindrical polar coordinates
and find the three corresponding Lagrange equations. (c) Describe the
solutions of the Lagrange equations in which ρ is constant.
6. What are the directions of the centrifugal and Coriolis forces on a
person moving: (a) south near the North Pole, (b) east on the equator,
and (c) south across the equator?
mα r′α = 0, where the position vector of the center of
P
7. Prove that
mass relative to the CM is zero. Solve for r′α and substitute into the
sum.
8. Consider the Atwood machine, but suppose that the pulley is a uni-
form disc of mass M and radius R. Using x as the generalized coor-
dinate, write down the Lagrangian, the generalized momentum p, and
the Hamiltonian H = pẋ − L. Find Hamilton’s equations and use them
to find the acceleration ẍ.
9. Consider a particle of mass m constrained to move on a frictionless
cylinder of radius R, given by ρ = R in cylindrical polar coordinates
(ρ, ϕ, z). The particle is subject to a force F = −krr̂. Using z and ϕ
as generalized coordinates, find the Hamiltonian H. Write down and
solve Hamilton’s equations and describe the motion.
10. Consider a system with one degree of freedom and Hamiltonian H =
2
H(q, p). Define new coordinates Q and P such that:
√ √
q = 2P sin Q, p = 2P cos Q.
(a)Prove that ∂H
∂q
= −ṗ and ∂H ∂p
= q̇, and show that the Hamiltonian
formalism applies to the new coordinates. (b) Show that the Hamil-
tonian of a one-dimensional harmonic oscillator with mass m = 1 and
force constant k = 1 is H = 12 (q 2 + p2 ). (c) Rewrite the Hamiltonian in
terms of Q and P and show that Q is ignorable. (d) Solve the Hamil-
tonian equation for Q(t) and verify that, when rewritten in terms of q,
the solution behaves as expected.
11. Find the Hamiltonian H for a mass m confined to the x-axis and subject
to a force Fx = −kx3 . Sketch and describe the phase-space orbits.
