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Introductory Physics Homework Help
Lagrangian Mechanics Problem
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[QUOTE="Yosty22, post: 5488846, member: 460241"] [h2]Homework Statement [/h2] 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 [h2]Homework Equations[/h2] Euler-Lagrange Equation [h2]The Attempt at a Solution[/h2] 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? [/QUOTE]
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Lagrangian Mechanics Problem
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