Double Pendulum and Normal Modes (Kibble problem)

[Simfish]

Homework Statement

1. A double pendulum, consisting of a pair, each of mass m and length
l, is released from rest with the pendulums displaced but in a straight
line. Find the displacements of the pendulums as functions of time.

===

So... this is a problem from Kibble's Classical Mechanics. Anyways, I can easily get the eigenvalues, eigenvectors, and normal modes for the double pendulum. But the problem is - I can't get the coefficients of the system unless I get a full set of initial conditions. Am I missing something? You need 4 sets of ICs to fully solve the approximation to this problem. The problem is -that you only know that the time derivatives (at time 0) are 0. As for the positions at time 0, all we know is that they're displaced in a straight line. But that just specifies one in relation to the other, and they could be fully horizontal. Or they could be displaced by any arbitrary angle...

 

[Nate810]

Homework Equations The equations of motion for the double pendulum:m\ddot{x_1} = -2ml\ddot{\theta_1}\sin\theta_1-ml\ddot{\theta_2}\sin(\theta_1-\theta_2)m\ddot{x_2} = -ml\ddot{\theta_1}\sin(\theta_1-\theta_2)-ml\ddot{\theta_2}\sin\theta_2l\ddot{\theta_1} = -g\sin\theta_1 + l\ddot{\theta_2}\cos(\theta_1-\theta_2)l\ddot{\theta_2} = -g\sin\theta_2 - l\ddot{\theta_1}\cos(\theta_1-\theta_2)The attempt at a solutionTo solve this problem, you need to first calculate the eigenvalues and eigenvectors of the differential equation. With these, you can construct the normal modes of the system. Then, you need to calculate the initial conditions (ICs) of the system. This is where my problem lies. You need 4 ICs to solve the system - two for each pendulum. The only thing I know to be true is that the time derivatives of both pendulums at time 0 are 0. But what about the positions at time 0? All I know is that they're displaced in a straight line. But that just specifies one in relation to the other, and they could be fully horizontal. Or they could be displaced by any arbitrary angle... Is there a way to get the exact positions of the pendulums at time 0 with only this information? Or is more data needed?