# Quantity conservation and calculus

1. Apr 28, 2009

### C.E

Hi, I was hoping someone could offer some guidance with the following, I don't even know how to start it.

The motion of a planet of mass m around the sun of mass M is governed by the following equation (note r is a vector):

d^2r = -k r (this should be the second derivative of vector r dt but I could not get it to underline)
dt^2 ||r||^3

where K=Gm (G is the gravitational constant) and r(t) is the position of the planet relative to the sun.

1. Show the following quantities are conserved.

a. Total energy (the sum of the potential and kinetic energies of the planet).

b. The angular momentum of the planet J.

c.The lenz- Runge vector L= dr x J - mk r
dt ||r|| (the dt should be under the dr and the ||r|| under the J - mkr )
Hint: (a x b) x c = (a.c)b- (b.c)a (note a.c is scalar product of a and c similarly b.c is scalar product of b and c).

2. (a) Interpret the constancy of J geometrically.
(b). Assume the planet moves in an ellipse with one focus at the sun, show by considering the point when the planet is furthest from the sun that L points in the direction of the major axis of the ellipse.