On reduction of two degrees of freedom Hamiltonian systems by an S1 action, and SO(1,2) as a dynamical group

Publisher Website
Google Scholar
Abstract
Reduction by an S1 action is a method of finding periodic solutions in Hamiltonian systems, which is known rather as the method of averaging. Such periodic solutions can be reconstructed as S1 orbits by pulling back the critical points of an associated ‘‘reduced Hamiltonian’’ on a ‘‘reduced phase space’’ along the reduction. For Hamiltonian systems of two degrees of freedom, a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two‐sphere in the Euclidean space R3, and the reduced Hamilton’s equations of motion are Euler’s equations. This article deals with the case where the reduced phase space will be a two‐hyperboloid in the three‐Minkowski space, and the reduced Hamilton’s equations of motion will be Euler’s equations with respect to the Lorentz metric. This reduction is associated with SU(1,1) symplectic action on the phase space R4. As a consequence of this association the reduced Hamiltonian system proves to admit a dynamical group SO0(1,2). A well‐known reduction by an S1 action occurs in the case of rotational‐invariant Hamiltonian systems, which will be associated with SL(2,R) symplectic action on R4. It is shown that the reduction associated with SU(1,1) and with SL(2,R) are symplectically equivalent.

This publication has 4 references indexed in Scilit: