Two Pendulums connected by a spring

[CAF123]

I get that $A = M^{-1}K$. I now have to diagonalise A. In my previous problems I have simply put the ansatz of $\underline{x} = \underline{P}(a \cos (wt) + b \sin (wt))$ and this has allowed me to obtain the form $G\underline{P} = \lambda \underline{P}$.

Is it what I should do here? It seems appropraite given the form of the energy and eqn of motion.

 

[voko]

Why do you need to diagonalize it? Are you supposed to find a closed-form solution?

 

[CAF123]

Why do you need to diagonalize it? Are you supposed to find a closed-form solution?

Yes, I am supposed to find a general solution to the equation of motion. This is probably a bit late to ask but what is the underlying physics behind diagonalising a matrix?

 

[voko]

The physics is mostly in eigenvalues and eigenvectors. These correspond to "pure tones", so to speak, which any complex motion of the system can be made of.

Diagonalization is a mathematical device that makes use of these physical elements to render the problem in a form easier to deal with. One could also say that it transforms the problem from the "ad hoc" coordinates that we used to describe the problem initially, into "intrinsic" or "physical" coordinates.

 

[CAF123]

The physics is mostly in eigenvalues and eigenvectors. These correspond to "pure tones", so to speak, which any complex motion of the system can be made of.

Diagonalization is a mathematical device that makes use of these physical elements to render the problem in a form easier to deal with. One could also say that it transforms the problem from the "ad hoc" coordinates that we used to describe the problem initially, into "intrinsic" or "physical" coordinates.

Thanks voko, I can take the rest of the question from here. Now to try the next problem - a double pendulum... :)