Can you help me with this mechanics question?

Posted on 2016-03-16 18:47:00 in Physics by TUFFEL

Two identical springs (with spring constant k and natural length 0) are joined in a 
loop and constrained to lie along the circumference of a circle of radius R = 1. Two identical masses m are attached to the springs, one at each junction. The positions of the masses are indicated by the coordinates q1 and q2. 
(a) Find the Lagrangian of the system (The system is assumed to lie in the x-y plane, so we don't have to consider gravity. 
I got the answer to be 0.5(m(q1')^2+m(q2')^2)-0.5k((q1)^2+((q2)... 
(b) Find the normal frequencies and the normal modes of the system. 
Because the Lagrangian is the way I found it I get only one normal frequency which i k/m giving a normal mode of (a,b)(a1cos(ω1t+γ)+b1sin(ω1t+γ)) where a and b are real numbers. 
(c) Assume that the system is initially at rest. For which initial values of the coordinates will the 
system oscillate about the positions q1 = 0, q2 = 0? Write down the corresponding solutions for 
q1(t) and q2(t). 
From the form of the potential energy I have that when q1=0 and q2=0 we have an equilibrium point but I don't know how to find the corresponding solutions for q1(t) and q2(t).

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