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## Homework Statement

We have a particle of mass m moving in a plane described by the following Lagrangian:

\frac{1}{2}m((\dot{x}^2)+(\dot{y}^2)+2(\alpha)(\dot{x})(\dot{y}))-\frac{1}{2}k(x^2+y^2+(\beta)xy) for k>0 is a spring constant and \alpha and \beta are time-independent.

Find the normal mode frequencies, \omega_1,2

## Homework Equations

Euler-Lagrange Equation

## The Attempt at a Solution

I think I'm just missing here. There was a lot of math, so I won't explicitly write out all of it, but I will have my final answers. I used the Euler-Lagrange Equation twice: once for x, once for y. This yielded:

m*ddot{x}+m\alpha(\ddot{y}=-kx+\beta(y)

and

m*\ddot{y}+m\alpha(\ddot{x}=-ky+\beta(x)

I solved each for \ddot{x} and equated them, giving me:

\ddot{y} = \frac{\beta+k/(\alpha)}{m(\alpha-1/(\alpha))}*y + \frac{-k-\beta/(\alpha)}{m(\alpha-1/(\alpha))}*x

Am I approaching this the right way to find the frequencies? I know usually in 1D for example, you solve the Euler-Lagrange equation to yield something of the form: \ddot{x}=\omega^2*x, but it is a little more unclear to me as to what to do here. Would I find two separate frequencies, once in x and once in y and they are two separate answers?