TY - JOUR

T1 - On Newton equations which are totally integrable at infinity

AU - Bialy, Misha

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2016/6

Y1 - 2016/6

N2 - 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.

AB - 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.

KW - 53C24

KW - Primary 37J50

UR - http://www.scopus.com/inward/record.url?scp=84971216472&partnerID=8YFLogxK

U2 - 10.1007/s00526-016-0985-8

DO - 10.1007/s00526-016-0985-8

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AN - SCOPUS:84971216472

SN - 0944-2669

VL - 55

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 3

M1 - 51

ER -