# Three-mass, two springs coupled oscillator NOT attached to walls

Int he three-mass coupled oscillator problem, we often see it stated that you have three masses, (they can be equal or not, but we'll assume they are equal here) connected by two springs and then another set of springs connecting the masses to the walls on the end. We assume that the spring constants are all the same. (We're assuming one-dimensional motion here as well).

So the equations of motion are usually written thus:

$$m\ddot x_1+kx_1-k(x_2-x_1)=0$$ $$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$ $$m\ddot x_3+k(x_3-x_2)+k(-x_3)=0$$

(And lord knows tell me if this is wrong and I missed a concept)

Well, ok, but what if we assume that the three masses are not connected to anything else? That is, they are connected by two springs but there are no walls? Would we just have to pick a reference frame and go from there, declaring one mas "stationary"? For instance, if we connect the first mass to a wall but leave the other end unconnected, would the EoMs look like:

$$m\ddot x_1+kx_1-k(x_2-x_1)=0$$ $$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$ $$m\ddot x_3+k(x_1+x_2)+k(x_3)=0$$

Anyhow, I was just curious, because all the problems in this vein I see seem to assume the three masses are attached between two walls.