Coupled oscillators and Normal Modes

Question

Consider we have a system consisting of 2 arbitrary masses and 3 arbitrary springs connecting them horizontally and between fixed walls, and we want to obtain the motion of each mass after we input some initial conditions ( but no driving force ).

After solving the two coupled differential equations that arose from the sittuation, we get that no matter which initial conditions we have, the chaotic motion of each mass is described simply by a linear combination of the two normal modes, with two different frequencies ( the normal modes happen to be sinusoidal because the restauring force is linear, making the ODEs linear, i suppose ).
But even though i can understand and follow all mathematical steps to derive such conclusion, i still lack a deep understanding and intuition into why the result holds.

So, I have two questions:

1 - Can anyone intuitively see why we can decompose such chaotic motion into a simple linear combination ?
Does this have anything to do with orthogonal basis of Function Spaces ( Vector space of functions ) ? I'm searching for a clear and simple mathematical or physical insight that allows us to understand that.

2 - Considering the previous question makes sense, is it true that the fact that the orthogonal basis is the cosine/sine one is due the restauring forces being linear ( and hence only the cosine/sine basis satisfies it ) ?

Thanks a lot in advance.

Answer 1

Because (as you say) the ODE is linear we have that if $\phi_i$ are all valid solutions then so is $$\sum_ia_i\phi_i$$ for any real $a_i$. You can convince yourself of this by substituting the sum into the ODE and showing it is satisfied assuming the $\phi_i$ are solutions.

We use the sin and cosine functions for our decomposition because they happen to form an orthonormal basis for the solution of the ODE in question. If we were solving a different (still linear) ODE then we might need a different set of functions. In particular we know that the sin and cosine functions form an orthonormal basis for periodic functions (by the Fourier series) so any (linear) ODE for which there are periodic solutions which form an ON basis can have it's solutions expressed as sinusoidal normal modes.

Answer 2

So you have your two normal modes, one is a sum, other is difference, and both of these change in time. If sum is zero, you have a difference and if difference is zero, you have a sum...any kind of motion can be pictured by mixing in some amount of sum and some amount of difference..in any position you have a sum of two amplitudes and a difference and they will change in a next moment and they certainly define position of two masses in some instant. That is how I visualize it.