Finding the Angles of Coupled Pendulums with Eigenvalue Analysis

[the_kid]

Homework Statement

I have three coupled pendulums: each of identical mass, and hung fromt the ceiling with identical massless rod. They are connected by identical massless springs. $\phi$$_{1}$, $\phi$$_{2}$, and $\phi$$_{3}$ represent their angles from the vertical, hanging position. Find these three angles as a function of time.

Homework Equations

The Attempt at a Solution

I started by setting up the Lagrangian, but I'm really not sure where to go from here. Any help?

 

[jncarter]

Have you drawn a picture? That should always be your first step with mechanics problems. Use the picture to label all the interactions and your generalized coordinates. Don't forget to use the interactions to find the relationship between the coordinates.
Taylor's "Classical Mechanics" goes through this problem in detail in the chapter on coupled oscillations.

 

[fluidistic]

I think he already has the Lagrangian. Assuming you didn't make any mistake, you'd have to write Euler-Lagrange equations using your Lagrangian. This will give you the equations of motion for each mass.

 

[jncarter]

Since you have an oscillatory system, your Lagrangian should be for the form $L = \frac{1}{2}\Sigma m_{ik}\dot{\phi}_{i}\dot{\phi}_{k} - \frac{1}{2}\Sigma k_{ik}\phi_{i}\phi_{k}$. So your equations of motion would be able to be written in matrix form. Is this what you have?

 

[the_kid]

Here is the caveat: at t=0, ϕ1=0, ϕ2=, and ϕ3=C.

I have the Lagrangian correct and I have the matrix form of the equation.

From there, how do I figure out the angles as a function of time, given those initial conditions?

 

[jncarter]

Solve the eigenvalue problem to find the frequencies (eigenvalues) and the amplitudes (eigenvectors). You can account for initial conditions by including an extra term in your general solution. Your solution should be something like $\phi_{i} = A_{i}cos(\omega*t - \delta)$ one of those terms can be used to account for the IC's (i indexes the coordinates 1,2,3; in other words $\phi$ is a "vector" you will have three such solutions, one for each eigenvalue $\omega_{\alpha}$). Which one and how would you solve for it?