Why do we assume that in normal modes, particles oscillate in form cos (wt) ?
How do we know that the general motion of particles can be expressed as a superposition of normal modes?
In both French and Crawford, the assumption of harmonic motion is made without any proof, please help.
In most cases, this is related to an assumption of small displacements from equilibrium. Assume that the system is described by a potential function $V(s)$, where $s$ represents the coordinate(s) associated with the normal modes. Let $s_0$ represent value of the coordinates the equilibrium state. Taylor expanding the potential about this point yields
$$ V(s-s_0) \approx V(s_0) + V'(s_0)(s-s_0) + (1/2) V''(s_0)(s-s_0)^2 + ... $$
The key feature is that we know $V'(s_0)=0$, since that is the the definition of equilibrium. We can also ignore the first term since it is independent of the state of the system.
Thus the resulting form of the equation of motion of the form
$$ 0 = \ddot{ s } + \omega^2 s^2 $$
with $\omega^2$ a function of $V''(s_0)$ and the masses/moments of inertia of the system. This equation has $\sin( \omega t), cos(\omega t)$ as its solutions.
Thus, simple harmonic motion is a generic feature of small oscillations about any mechanical equilibrium.
