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Which condition(s) should the energy satisfy, such that the solution of the corresponding Euler-Lagrange equation is oscillating?

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  • $\begingroup$ Can you please explain what Euler-Lagrange equation you are considering? $\endgroup$
    – fra_pero
    Commented May 13, 2020 at 13:11
  • $\begingroup$ I do not have a special Euler-Lagrange equation in mind (so not only harmonic oscillation). I am searching for a general consideration. $\endgroup$
    – Q.stion
    Commented May 13, 2020 at 15:24

1 Answer 1

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Non-linear systems of odinary differential equations are pretty well studied. In particular, a conservative system with an arbitrary potential energy (so-called nonlinear oscillator) will have only one type of stable states, corresponding to its potential minina, with periodic oscillations around each of them.

There may be also periodic trajectories in more complex systems with energy sources and dissipation, but these are probably beyond of what you are asking.

Unfortunately, I am unable to recommend any good books in English. Here is my list of basic Russian sources (mist existing in translation).

If course, one can also have oscillations in Euler-Lagrange equations without any potential at all - with magnetic field.

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