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On Newton equations which are totally integrable at infinity

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Abstract

In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions 3 and higher the following rigidity results holds true: if all the orbits in a neighborhood of infinity are action minimizing then the potential must be constant. This gives a generalization of the previous result Bialy and Polterovich (Math Res Lett 2(6):695–700, 1995), where it was required all the orbits to be minimal. As a result we have the following application: suppose that for the time-1 map of the Hamiltonian flow there exists a neighborhood of infinity which is filled by invariant Lagrangian tori homologous to the zero section. Then the potential must be constant. Remarkably, the statement is false for \(n=1\) case and remains unknown to the author for \(n=2\).

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Authors and Affiliations

  1. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

    Misha Bialy

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  1. Misha Bialy
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Correspondence to Misha Bialy.

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The author declare that he has no conflict of interest.

Additional information

Communicated by P. Rabinowitz.

Partially supported by Israel Science Foundation Grant 162/15.

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Bialy, M. On Newton equations which are totally integrable at infinity. Calc. Var. 55, 51 (2016). https://doi.org/10.1007/s00526-016-0985-8

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  • DOI: https://doi.org/10.1007/s00526-016-0985-8

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