# Classical Mechanics problem

1. Feb 2, 2008

### neelakash

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

Last edited: Feb 2, 2008
2. Feb 2, 2008

### 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.

Last edited: Feb 2, 2008