Question 1: In cylindrical coordinates (ds)2 = (dΡ)2 + p2(dΦ)2+ (dz)2. Find the non-zero Christoffel symbols and the equations, of the geodesic, for the two dimensional case of move-ment on the surface of a cylinder of radius Ρ = R.
Question 2: The motion of an object of mass m in two dimensions, in the presence of a gravita¬tional field may be determined from the Lagrangian
L = 1/2m (x2 + y2) - mgy
Consider the transformation given by
x' = x + α ± βt
y' = y
where α and β are constants. For this system
(a) Show that the transformation is an invariance transformation by finding the function A.
(b) Find the related infinitesimal invariance transformation and calculate SA.
(c) Find the associated constants of the motion for the two parameter transformation above.