# How is energy distributed over the normal modes?

Consider a linear chain of $$N$$ point masses $$m$$ connected by linear springs $$k$$ and fixed at the two ends by rigid walls separated by a distance $$L=N\times l$$.

If I take the point mass at the center $$m_{i}=m_{\frac{N}{2}}$$ and pull it by small distance $$\delta x$$ to the right, and then release, then

(1) How is energy distributed over the normal modes ?

(2) Does this distribution change if energy transfer between modes is allowed via non-linear instead of linear springs ?

(1): if you solve the linear eigenvalue system of normal modes, you can construct the most general solution as $$f \propto \sum_j c_j o_j(t) d_j(x)$$ where $$o_j(t)$$ is the time dependence of mode $$j$$ and $$d_j(x)$$ is the spatial dependence. Then at $$t=0$$ you just do the Fourier transform and find the coefficients $$c_j$$ that depends on the initial condition distribution. Then, the energy distribution is basically $$|c_j|^2$$ (normalized such that the sum equals the initial energy content).
(2): if you have non-linear terms in the equation of motion, simply the $$c_j$$ will be functions of time as well $$c_j(t)$$, and there will be a "statistical" steady-state, time-averaged, for which the average $$c_j$$ distribution could be different than the linear one for sure (but what exactly you end up with, depends on the type and strength of non-linear couplings).
• Yes this would simple standard methodology. But I am looking in literature and here for established qualitative knowledge about the expected behaviour before formalising and calculating the problem. For example what sort of dependence should be expected of $k$ and $m$ ? Commented Dec 10, 2023 at 10:21