Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very degenerate, because all the energy levels with close energy values do not contain periodic trajectories: a really weird picture!
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2 Answers
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If we take $H=I + \frac13(p^3 -q^3) + I^2(p-q)$ with $\omega =dI\wedge d\theta + dp\wedge dq$,
then
$\dot q = p^2 + I^2$
$\dot p = q^2 + I^2$
$\dot I = 0$
$\dot\theta = 1 + 2I(p-q)$
and the only periodic orbit is for $q=p=I=0$.
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$\begingroup$ A very nice example! What is interesting is that this system is also integrable (the second integral is $I$) and reversible with respect to the phase space involution $G:(q,p,I,\theta)\mapsto(-p,-q,I,-\theta)$, and the cycle $\{q=p=I=0\}$ is invariant under $G$. $\endgroup$ Commented Jan 4, 2018 at 14:27
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$\begingroup$ Glad you liked it! As you point out it is integrable, it shows too a difference between compact and non-compact energy levels of integrable systems (for compact levels Liouville-Arnold implies the periodic orbits come in families). $\endgroup$ Commented Jan 4, 2018 at 18:27
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$\begingroup$ No, compactness does not matter here. Assume each of the variables $I$, $p$, $q$ to range in $[0;2\pi]$ (like $\theta$) and take $H=\sin I+\frac{1}{3}(\sin^3p-\sin^3q)+\sin^2I(\sin p-\sin q)$ with the same $\omega$, then again $I$ is another integral and $q=p=I=0$ is a periodic orbit which is no longer the only periodic orbit whatsoever, but the only periodic orbit in its neighborhood, although now the whole phase space is compact (the $4$-torus). The reason is that $dH=dI$ along the orbit. By the way, the new system is still reversible with respect to the same involution $G$. $\endgroup$ Commented Jan 6, 2018 at 17:37
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$\begingroup$ Ah, I see thanks. It is just that the integrals are dependent along the orbit. $\endgroup$ Commented Jan 9, 2018 at 1:36
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Another example and many generalizations are presented in the paper
Mikhail B. Sevryuk, Integrable Hamiltonian systems with a periodic orbit or invariant torus unique in the whole phase space, arXiv:1808.03596
available at http://arxiv.org/abs/1808.03596.
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$\begingroup$ See also a full-text view-only version of this paper just published "online first" in Arnold Mathematical Journal: link.springer.com/epdf/10.1007/… $\endgroup$ Commented Nov 23, 2018 at 10:10
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$\begingroup$ Here is the complete reference: Arnold Mathematical Journal, 2018, V. 4, No. 3-4, pp. 415-422. $\endgroup$ Commented May 15, 2019 at 18:59