Assignment:
The energy and angular momentum of a particle inside a central potential are given by:
E = ½*μ *(dr/dt) + V(r) L = μ* r^2 *(dθ/dt)^2
V(r) = (G*μ*M)/r + L^2/(2*μ*r^2)
a) Solve these two equations for dr/dt and dθ/dt and show that:
dr/dt = +/- [ 2/(μ*(E - V(r)))]^1/2 dθ/dt = L/(μ*r^2)
b) Write the equation of a) in a μ independent form by introducing e = E/μ and l = L/μ
c) Use units such that G = M = c = 1, and let e = -.02, l = 4.5. Let the initial position be given by let the initial position be given by r(t=0) = a, θ(t=0) = 0
a = -GM/(2E). Compute the initial velocities dr/dt(t=0) and dθ/dt(t=0) under the assumption both are positive.
d) Plot V(r)