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1. (a) Show that if in a variational problem, to extremise
∫ (^) x 1
x 0
F (y, yx, x)dx,
it is given that ∂F ∂x
then the ’energy’ E: E = yx
∂F
∂yx
− F
is constant if y satisfies the Euler-Lagrange equation.
(b) A surface of revolution S is expressed in cylindrical polar coordinates (r, θ, z) as
r = f (z), −b ≤ z ≤ b.
Such a surface intersects the planes z = ±b in circles r = a. You may assume that f (z) > 0 for all z ∈ [−b, b]. Write down, as an integral with respect to z, the area A of the surface S. If the surface is such as to minimise A, write down the Euler-Lagrange equation which f (z) must satisfy. Show that this Euler-Lagrange equation has a first integral,
g(f, fz ) = constant.
Find this integral, and hence show that the general solution of this Euler-Lagrange equation is f (z) =
k cosh(kz − l), where k and l are arbitrary constants. Find the solution satisfying the given boundary conditions, r = a at z = ±b. Show, graphically or otherwise, that there is no solution to the problem if b/a is too large.
2. N particles of masses mi, and position vectors xi , i = 1,... , N , move in the plane,
interacting pairwise via potentials Vij (|xi − xj |). There is no external potential.
(a) Write down the Lagrangian of the system. (b) Describe the symmetries of the system, explaining carefully what you mean by a symmetry. Hence construct 5 integrals of motion for the system. (c) Show that after a simple change of coordinates, three of these integrals may be fixed to be zero. (d) In the planar 3-body problem, N = 3, and Vij = − Gmimj |xi − xj |
Given that no other integrals of motion are known in this case, discuss briefly whether or not the system is integrable.
3. A rod is inclined at an angle α to the horizontal. A mass m is free to slide along it. A mass
m hangs from the first mass, on a string of length l. Gravity acts vertically downwards.
(a) Show that the Lagrangian L for the system, using coordinates as in the diagram, is given by:
L = m X˙^2 + ml X˙ θ˙ cos(θ − α) + m 2 l^2 θ˙^2 + mgl cos(θ) − 2 mgX sin(α).
(b) Derive the two Euler-Lagrange equations for the system. (c) Calculate the energy E of the system, and hence write down the Hamiltonian. (d) Show that translation in X is a symmetry of L; hence or otherwise construct a second integral of motion.
5. A point mass m with position vector x rotates about the origin with angular velocity vector
ω.
(a) Write down the kinetic energy of the particle in the form
T =
ωT^ Iω,
giving the 3 × 3 matrix I, the inertia tensor, explicitly. Hence write down the inertia tensor for a system of N rigidly connected point masses, mi at points xi, satisfying |xi − xj | = constant. What is the angular momentum of the system if the angular velocity is ω? (b) State Euler’s equations for a rotating body. What relation do the inertia tensor, angular velocity and angular momentum defined in part (a) of the question have with the analogous quantities appearing in Euler’s equations? (c) A body consists of 2 masses M and 2 masses m, with M > m, rigidly fixed to a light square frame as shown in the diagram
Write down its inertia tensor in the (X 1 , X 2 , X 3 ) coordinates. Write down Euler’s equations for this system, state what steady rotations are possible, and state, with reasons, which ones are stable.
