Question

Given L, find the form of V (r) so that the path of a particle is given by the spiral r = Cθ^k, where C and k are constants. Hint: Obtain an expression for r ̇ that contains no θ’s

Answer 1

You need to be more specific... I'm not sure what V and L are representing.

Answer 2

it has to do with the lagrangian method. Specifically Central forces. Basically you have the kinetic energy and potential where the V(r) in this case is the potential. The L is i think the angular momentum so Lagrangian = m*rdot/2 + L^2/2mr - V(r) or something in that direction but I dont know exactly what to do hence im posting here :)

Answer 3

Does it have to be true for all energy E? If not, for some energy E, you can write T = E - V, and replace rdot = dr/dθ * dθ/dt and angular energy 1/2 m r^2 θdot^2 in T with θdot = L/mr^2, then put the r = C θ^k into dr/dθ and eliminate θ. Then you get an equation with only r dependence, and a free V(r) function to manipulate.