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.

Original languageEnglish
Article number51
JournalCalculus of Variations and Partial Differential Equations
Volume55
Issue number3
DOIs
StatePublished - Jun 2016

Keywords

  • 53C24
  • Primary 37J50

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