ref: fe6b680fd367b85811a3d1400efa48b781ad6501
physics-notes/md/small-oscillations.md
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We consider a mechanical system near a stable equilibrium.

A stable equilibrium at a position $q_0$ is where the potential $U(q)$ is a local minimum at $U(q_0)$. A movement away from this position leads to setting up a force $-\dd{U}/\dd{q}$ which tends to return the system to equilibrium. We choose a co-ordinate system where the equilibrium corresponds to $x=0$

```
LL1/21.1
```

We put $U(q_0)=0$ as a base energy We put $U'(q_0)=0$ because we don't consider asymetrical potentials We note $U''(q_0)=k\neq 0$ because we don't consider potential of higher order. Then consider the series expansion of $U(q-q_0)$, and keeping the lowest terms for small deviations of equilibrium we have

```
LL1/21.2
```

The kinetic energy, with one degree of freedom is of the form $\mfrac{1}{2}a(q)\dot{q}^2=\mfrac{1}{2}a(q)\dot{x}^2$. In the same approximation as above, $a(q)=a(q_0)$.

We note $a(q_0)=m$, this is the mass only if $x$ is the Cartesian co-ordinate.

```
LL1/21.3
```

Using `LL1/2.6`

we derive the equation of motion, which is called the harmonic oscillator

```
LL1/21.5
```

where

```
LL1/21.6
```

The general solution the harmonic oscillator is

```
LL1/21.11
```

Where $A=ae^{i\alpha}$ is the complex amplitude, composed of the real amplitude $a$ and the phase $\alpha$ which depend on the initial condition of the system $x(0)$ and $\dot{x}(0)$.

We note that the frequency $\omega$ doesn't depend on the inital condition, but only on the parameters of the system $k$ and $m$.

We also note, that the frequency of the motion is independant on the amplitude, which we have already predicted with `LL1/10.2`

for a quadratic potential $k=2$.