# Two Dimensional Oscillators

Potential

## Isotropic Oscillator

The simplest 2-D oscillator that is composed of identical springs (same spring constant).

The equations of motion are separable two equations, one for each direction

you could define a relative phase between the two oscillators as

it can be substituted into the above equations by shifting the time origin (problem 5.15)

let

then

let

similarly

## Interpretation

There are several modes of motion for the above system of equation

### No phase difference

If there is no phase difference then (\delta =0 ) and you have oscillating motion along a line.

### 90 degree phase difference

If the x and y motions are completely out of phase ( phase difference of 90 degrees or )

then the motion is an ellipse (unless their amplitudes are equal in which case the mass moves in a circle).

The above is an equation of a circle if otherwise it is an ellipse with semi-major and semi-minor axes and

If the phase shift is less than 90 then the ellipse is slanted towards the x-axis otherwise it is toward the y-axis.

## Anisotropic Oscillator

Consider the case of having two different spring constants in each direction

The equations of motion are separable two equations, one for each direction

where

### Lissajous figure

if

rational number (a fraction formed two integers, denominator is non-zero)

then the motion repeats itself.

If the ratio is irrational the motion does not repreat.

problem 5.17 a.) show motion is periodic if the ratio of frequencies is a rational number ( p/q)

Assume that the ratio of oscillation frequencies is

where p & q are integers

The period of the oscillation will be given by

Definition of period

The solution for the next period is

Motion is periodic
if p is an integer

Similarly for the y-direction

Motion is periodic
problem 5.17 b.) show motion is not periodic (no repetition) if the ratio of frequencies is a IR-rational number

For this solution you just assume the motion is periodic and start the solution to part 5.17 a.) from the end and work backwards.