Solve Coupled Oscillator Problem from Goldstein's Classical Mechanics

[physicsjock]

Hey,

I've been trying to solve this question from Goldstein's Classical Mechanics.

The picture I have of the question is from a later edition and the hint was removed from the question, the hint was let
η₃=ζ₃

η₁=\frac{ζ_{1}+ζ_{5}}{\sqrt{2}}

η₅=\frac{ζ_{1}-ζ_{5}}{\sqrt{2}}

What I have done is first let each particle be represented by a displacement x1...x5,

Then wrote out T = 1/2m(x^{2}_{1}+x^{2}_{3}+x^{2}_{5}) + 1/2M(x^{2}_{2}+x^{2}_{4})

and V = k/2 *( x_{i}-x_{j}-b ) i = 2..5, j = 1..4 i≠j

so V = k/2 *( x_{2}-x_{1}-b ) + k/2(...) up to i = 5 j = 4

then Since η = x - dx the system is at equilibrium when

b = dx₂ - dx₁ = dx₃ - dx₂ = ... up to i = 5 j = 4then V = 1/2k (η₂ - η₁) + ... up to i = 5 j = 4

Then I subbed in the hints it provided and also as one of the hints says treat the normal co-ords of 2 and 4 as symetric I let η₂=ζ₂=-η₄

Some stuff canceled and I ended up with

V = k/2 *( ζ^{2}_{1}+ζ^{2}_{2}+ζ^{2}_{5}-2\sqrt{2}ζ_{2}ζ_{5} )

I turned it into a matrix which was (this is where I start stuffing up I think)

...1...0...0 (sorry had to use the ... to make the matrix look kind of like a matrix)
V = k/2...0...4...sqrt2
...0..-sqrt2..1

Then since there were only varibles of 1, 2 and 5 I turned T into

.....m 0 0
T = 1/2...0 M 0
.....0 0 m

Then did the usual thing for eigenvalues |V-ω²T|=0

One was pretty ugly, one was sqrt(k/m), and the last one I had trouble finding because it was a mess of a cubic.

I decided to put the question at the bottom so the add didn't squish it,

http://img193.imageshack.us/img193/7747/asdasddh.jpg

Is what I did alright? The part I'm not confident about at all is when i turn V into a matrix, and the book they also drop the 1/2 for both V and T which I didn't really understand why.

Thanks in advanced for any help.

 

[physicsjock]

So one problem was that symetric meant n2=n4, so i did it again,
its a 5 variable quadratic form so its a 5x5 not a 3x3,
I got a block diagnal matrix with a 3x3 as the first and a 2x2 as the second, the 2x2 had determinant kmw^2, so that shows one of the frequencies is 0,

but yea I am still not sure if that's right,

Anyone have any ideas?