PC4274A Assignment 1

Question 1

The Schwarzschild metric for the static field of a non-rotating spherically symmetric black hole of mass M is given by

where t is the time coordinate, (r, θ, φ) are spherical-like coordinates, G is Newton’s gravitational constant and c is the speed of light. The path of a small test particle in this field can be described in term of proper time τ , i.e. (t(τ ), r(τ ), θ(τ ), φ(τ )).

(a) Considering only the motion confined to the so called equatorial plane θ(τ ) = π/2 and as- suming that the path of the small test particle is such as to make
ds stationary, find two first integrals of the equations of motion.

(b) From their Newtonian limits, in which GM/r, ̇r2 and r2 ̇φ2 are all ≪ c2 where ̇r ≡ dr/dτ and ̇φ ≡ dφ/dτ , identify the constants of integration.

Question 2

(a) We seek to extremize the functional

with respect to functions which attain the value u1 for x = x1 and which satisfy the given relation g(x, u) = 0 at the upper limit of integration, as yet undetermined. Show that the stationary function u(x) satisfies the Euler equation

and, in addition to the left-hand end-point requirement u(x1) = u1, the right-hand end-point condition

(b) Find the shortest distance between the line f (x) = x − 3 and the curve g(x) = ex. Also, identify the point on respective curves giving the shortest distance.

Question 3

Determine the stationary functions u(x) and v(x) for the functional

subject to the boundary conditions

and the constraint

Question 4

(a) The Lagrangian for a particle with charge q and mass m in an electromagnetic field described by electric scalar potential φ(r(t), t) and magnetic vector potential A(r(t), t) is

Obtain the equation of motion of the charged particle from Hamilton’s principle.

(b) The Lagrangian density of an electromagnetic field with a charge density ρ(r, t) and current density J(r, t) is given by

The Lagrangian of an electromagnetic field is given by a volume integral of the Lagrangian density: