- #1
fluidistic
Gold Member
- 3,926
- 262
Homework Statement
I'm stuck at understanding how to find the kinetic and potential energy matrices such that the determinant [itex]|V- \omega ^2 T|=0[/itex] when solved for [itex]\omega[/itex], gives the normal modes (characteristic frequencies?) of the considered system.
For example in Goldstein's book for a molecule of the type m---M---m where the masses are linked via springs of constant k, [itex]V=\frac{k}{2} (\nu ^2 _1+2 \nu ^2 _2 + \nu ^2 _3 -2 \nu _1 \nu _2 - 2 \nu _2 \nu _3)[/itex] where [itex]\nu _i =x_i-x_{0i}[/itex].
He then says "hence the V matrix has the form [itex]\begin{bmatrix} k & -k & 0 \\ -k & 2k & -k \\ 0 & -k & k \end{bmatrix}[/itex]". I notice it's symmetric as it should, since I guess it has been diagonalized.
He continues on by saying that the kinetic energy [itex]T=\frac{m}{2}( \dot \nu ^2 _1+ \dot \nu ^2 _3)+\frac{M}{2} \dot \nu ^2 _2[/itex]. I've no problem understanding this, but I do have a problem when he says that thus the T matrix is diagonal and worth [itex]\begin{bmatrix} m & 0 & 0 \\ 0 & M & 0 \\ 0 & 0 & m \end{bmatrix}[/itex]. How did he get this matrix?!
When I look for the definitions of both the T and V matrix, it's even more confusing to me.
[itex]T=\frac{1}{2} \dot \Xi ^t \dot \Xi[/itex]. And it's not even clear to me what is the definition of the [itex]\Xi[/itex] matrix. It seems it has to see with coordinates but I don't understand anything else than this. While [itex]V=\frac{1}{2} \Xi ^t \lambda \Xi[/itex]. Where the ^t mean transpose (yes, they are matrices).
So I'm given the following problem:
Find the normal modes of longitudinal vibrations of the chain of masses like this: Wall-k-m-k-m-k-m-k-Wall. Where the "-" are springs.
Homework Equations
Not sure.
The Attempt at a Solution
I feel I can't do my problem if I don't understand how to calculate the matrices V and T from the expression of V and T. I do understand how to get V and T (scalars), but not their link to the T and V matrices.