Finding the Lagrangian Matrix for Two-Spring Systems

[MattIverson]

Homework Statement

The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system).
I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm not sure what the n's are and what they represent, so I'm worried that I'm missing something. I also am not sure about the equillibrium position...
Then I'm pretty sure I can do part (b) and (c), but I'm wondering if I just find the eigenvector and values like a normal linear algebra problem. Do I simply just take the matrix I have and operate on it?

Homework Equations

L=T-V
e^-iwt
d/dt(dL/dqdot)-(dL/dq)

The Attempt at a Solution

Attached (for the second system). I'm pretty sure I can figure out (1) if I figure out (2).

 

[vela]

You didn't do what was asked for in part (a). It's asking you to find two matrices, $\mathbf T$ and $\mathbf V$, for one thing.

In the Lagrangian, you have for the kinetic energy $T=\frac 12 m(\dot x_1^2 + \dot x_2^2)$. You can write this as
$$T = \frac 12 \begin{pmatrix}\dot x_1 & \dot x_2 \end{pmatrix} \begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix}\begin{pmatrix}\dot x_1 \\ \dot x_2 \end{pmatrix}.$$ The matrix of masses is the matrix $\mathbf T$ you're being asked to find. You can express the potential energy similarly in terms of a matrix.