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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 ?

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(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).

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  • $\begingroup$ 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 $m$ ? $\endgroup$ Commented Dec 10, 2023 at 10:21
  • $\begingroup$ math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – hyportnex
    Commented Dec 10, 2023 at 12:02
  • $\begingroup$ yes, I just got how to use it. will do from next one! Btw thanks a lot for having edited mine! $\endgroup$
    – drgetwrekt
    Commented Dec 10, 2023 at 12:05
  • $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented Dec 10, 2023 at 12:41
  • $\begingroup$ The community bot seems to have closed my question because of bad formatting in this answer, great.. $\endgroup$ Commented Dec 10, 2023 at 15:27

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