Is the Eigenvalue Equation Correct for 3D Small Oscillations?

[neelakash]

Homework Statement

Problem: Derive eigenvalue equation of motion of a system undergoing free small oscillation in three dimensional space.

Homework Equations

The Attempt at a Solution

I want to know if I have correctly written the values of T and V---

\ T=\frac{1}{2}\ m [\dot{x}^2 +\dot{y}^2+ \dot{z}^2]

And \ V=\frac{1}{2}\ k[\ x^2+\ y^2+\ z^2]

I m not sure if I have written the write thing...what should be the sign of k.Should I write -ve?And is it OK to assume that the force constants are equal?What would be if they were not equal?Would they simply add like (1/2)(k1+k2) x^2?

If I can form T and V,I am confident that I will be able to do the rest of calculations.

 

[neelakash]

It appears to me that the potential energy would be

\ V=\frac{1}{2}[\ k_1+k_2]\ x^2+...

From the elimentary knowledge, we know when a spring of natural length is compressed or elongated, workdone on the particle is -(1/2)kx^2. So,...(1/2)kx^2 amount of energy is stored within the spring as potential energy.So, V would be +ve.

Right?

So, we are to omit the factor of half in V in the first post. Please tell me if I am correct.