Two Dimensional Oscillators

Isotropic Oscillator

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

[math]\displaystyle{ \vec F = -k( x \hat i + y \hat j) }[/math]

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

[math]\displaystyle{ \ddot x = - \omega^2x \Rightarrow x = A_x \cos(\omega t - \delta_x) }[/math]

[math]\displaystyle{ \ddot y = - \omega^2 y \Rightarrow y = A_y \cos(\omega t - \delta_y) }[/math]

you could define a relative phase between the two oscillators as

\delta = \delta_y - \delta_x

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

let

[math]\displaystyle{ t^{\prime} = t + t_0 }[/math]

then

[math]\displaystyle{ x = A_x \cos(\omega t^{\prime} - \omega t_0 - \delta_x) }[/math]

let

[math]\displaystyle{ t_0 = \frac{-\delta_x}{\omega} }[/math]

[math]\displaystyle{ x=A_x \cos(\omega t^{\prime} - \omega\frac{-\delta_x}{\omega} - \delta_x) }[/math]

[math]\displaystyle{ =A_x \cos(\omega t^{\prime}) }[/math]

similarly

[math]\displaystyle{ y = A_y \cos(\omega t^{\prime} - \omega t_0 - \delta_x) }[/math]

[math]\displaystyle{ =A_y \cos(\omega t^{\prime} - \omega \frac{-\delta_x}{\omega} - \delta_x) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} +\delta_x - \delta_y) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} +\delta) }[/math]