Two rotating masses attached by a spring

1. The problem statement, all variables and given/known data



Two pucks of mass m slide freely on a horizontal plane. They are connected by a

spring (constant k and negligible un-stretched length) and set in circular motion

with angular momentum L. The pucks are given a small, simultaneous radial

poke. What is the frequency of subsequent radial oscillations?



2. Relevant equations



Lagrangian? -> Equations of motion?



3. The attempt at a solution

I thought of setting up the Lagrangian of the system and finding the Equations of Motion and then some how apply the small radial pertabation and from that the radial frequency should just pop out. My Lagrangian is

L=1/2m(r1'^2+r2'^2)+1/2m((r1*θ1')^2+(r2*θ2')^2)-1/2k(r1-r2)^2. Since the system is set in circular motion all θ' are constant such that θ'=ω for both theta. With that said the equations of Motions are

-m*r1''+m*w^2*r1-k(r1-r2)=0

&

-mr2''+m*w^2*r2+k(r1-r2)=0

Now how apply the small radial perturbation? is r1-r2<<r1 or r2?

(Note that all time derivatives are denoted by a ', i.e. v(t)=x'(t))