- #1

- 49

- 0

## Homework Statement

A linear chain consists of N identical particles of mass m are connected by N+1 identical, massless springs with force constant

*k*. The endpoints are fixed to walls on each side. In the static configuration each spring is stretched from its relaxed length

*l*

_{0}to a new length

*l*. Construct the Lagrangian for general small oscillations in a plane and show that the transverse and longitudinal modes decouple. Discuss how the frequencies depend on

*l*-

*l*

_{0}. Discuss and interpret the behavior for

*l*<

*l*

_{0}.

## Homework Equations

[itex] T = \frac{1}{2} \Sigma m_{\alpha} v^{2}_{\alpha} [/itex]

[itex] U = \frac{1}{2}\textit{k}(\textit{l}-\textit{l}_{0})^{2} [/itex] I think

L = T - U

## The Attempt at a Solution

I'm having a hard time deciding on the best way to define my coordinates and how they all relate to each other. I know that once I have that figured out I must assume that the x and y displacements are small and use a Taylor expansion. If there are no cross terms or x and y, then the longitudinal and transverse waves must be decoupled. I think I just need the right picture to be drawn.

Thanks for all of the help!