~jzck/physics-notes

ref: 508d894fe48a45d461c3b675e597cedb89692f79 physics-notes/md/small-oscillations.md -rw-r--r-- 1.6 KiB
508d894fJack Halford small oscillations 1 year, 2 months ago

#type: model title: small oscillations

We consider a mechanical system near a stable equilibrium.

#In one dimension

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 $m$.