Equation of motion for coupled oscillators

[snellslaw]

Attached is the formula for the equation of motion for a system of coupled oscillators.
Could someone please tell me what the variables and indices refer to here?
Thanks! :)

 

[tiny-tim]

hi snellslaw!

instead of the easy case of the force on an oscillator being proportional to its displacement,

it's a function of the displacements of itself and another oscillator

 

[snellslaw]

Thanks TinyTim! So is "a" the spring constant?

 

[tiny-tim]

there are n² a's

 

[mps]

So are the "a"s just constants?
x is the displacement from equilibrium right?

Thanks! :)

 

[tiny-tim]

yes, the n x's are the n displacements, and the n² a's are the n² (or n²/2 if they're symmetric) constants

 

[AbigailM]

The i means you are dealing with a set of equation. a is a coefficient and summation symbol means that each equation has a certain number of terms. For example:

$\ddot{x_{1}}=a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}$

$\ddot{x_{2}}=a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}$

$\ddot{x_{3}}=a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}$

Where i = 1 to 3 and j = 1 to 3. You have 3 equations with 3 unknown, and can solve this using the matrix determinant method.

Regards.

 

[snellslaw]

@mps: hey are you preparing for the same thing? :P

Thanks everyone!