I'm studying Hamiltonian system and in particular the role of frequencies in these systems. What I want to understand is about the physic interpretation of some definition.
Considerer a Hamiltonian system of the form $$ \sum_{l=1,...,n} \dfrac{y_l^2}{2} + V(x) $$ where $(x,y)$ are canonical variables and $V$ is the potential. Assume that $V$ has a minimum at the origin. Then by standar theory there exist a canonical variables (normal modes) $(\bar p, \bar q)$ in wich the Hamiltonian takes the form $$\sum_{l=1,...,n} \dfrac{\bar p_l^2 + \omega_l ^2 \bar q_l^2}{2} + P(\bar q)$$ Thus if one neglect the function $P$ one can consider the system as a system of $n$ independent harmonic oscillators with frequencies $w_l$.
I can interpret all using a chain of $N$ masses attached with springs. In this way I can image the dynamic of the whole system as the superposition of normal mode, and I can visualize the motion of normal modes and what the frequencies represent in the motion.
Now define non resonant frequencies: given the vector of frequencies $\omega=(\omega_1,..., \omega_n)$ the frequencies are nonresonant if $<\omega,k>\neq 0 $ for every $k \in \mathbb{Z}-\{0\}$.
My question is: what does it change in the motion of masses when one consider non resonant frequencies and the completely resonant one ( for example, fixed $\omega$, $\omega_l= \omega$ for every $l$)? In which sense there is "an exchange of energy among the oscillators in the resonant case and how do the masses actually move?