Continuation of Periodic Orbits in Symmetric Hamiltonian Systems
The idea of everything returning eventually to its point of departure has a strong hold on humanity, with many historical, philosophical and religious implications. Classical examples are the need to construct a calendar and the subsequent search for orbits in the solar system in which the planets follow a closed track and repeat their history over and over again.
“It seems at first that the existence of periodic solutions could not be of any practical interest whatsoever. Indeed, the probability is zero for the initial condition to correspond precisely to those of a periodic solution. But it may happen that they differ by very little. […] Here is a fact which I have not been able to demonstrate rigorously, but which nevertheless seems very plausible to me. Given equations of the Hamiltonian form and any particular solution of these equations, we can always find a periodic solution (whose period may admittedly be very long) such that the difference between the two solutions is as small as we wish during as long a time as we wish. Besides this, what renders these periodic solutions so precious is that they are, so to speak, the only opening through which we may try to penetrate into the fortress which has the reputation of being impregnable.”
KeywordsPeriodic Solution Periodic Orbit Hamiltonian System Bifurcation Diagram Relative Equilibrium
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