# Oscillators¶

In general, the Lagragian for a system with n general coordinates can be

To write down equation of motion, we need the following terms,

Then equation of motion is

Generally, we can’t solve this system. But there is an interesting limit. The system may have equilibrium points. We can study systems oscillating around equilibrium points.

At equilibrium, the system can stay steady, i.e., \(\dot q_j^0 = 0\). This gives us

for all j.

Now for small deviations, we can expand the system around equilibrium points.

Then

So we have the Lagrangian for small oscillations,

Typing indices using LaTeX is so annoying. So we’ll use matrix notations and Lagragian becomes

in which \(T\) and \(V\) matrices are n by n real and symmetric.

(We need to diagonalize T and V. First question comes to us is:

**Is is possible to diagonalize both T and V at the same time?**

We can have a look at the surface \(\tilde p T p = C\), which is a elliptical surface with coordinates \(p\).)

Use the following transformation

Then transpose

So we have the new Lagragian

Define \(T^{-1/2} V T^{-1/2} \equiv V'\).

Next we need to diagonalize V’ by using its eigen vectors.

is equivalent to

with \(b = T^{1/2} a\). So we have

is same as

in which \(\lambda\) is the eigen value of this function.

## Simplest Harmonic Oscillators¶

Harmonic oscillators are described by

which has solution

where \(\omega = \pm \sqrt{ \frac{k}{m} }\) and the final solution is determined by the second initial condition, i.e., the first order derivative of displacement.