Abstract
It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase space and with uniqueness replaced by isolatedness are also constructed. Finally, reversible analogues of all the examples are described.
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Acknowledgements
The author is grateful to L. M. Lerman for useful discussions on the role of compactness of various invariant manifolds in the phenomena studied in this note.
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Sevryuk, M.B. Integrable Hamiltonian Systems with a Periodic Orbit or Invariant Torus Unique in the Whole Phase Space. Arnold Math J. 4, 415–422 (2018). https://doi.org/10.1007/s40598-018-0093-2
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DOI: https://doi.org/10.1007/s40598-018-0093-2