2010/11/26 by Charles-Michel Marle, Marle, Charles-Michel
Mathematics · Physics and Astronomy · #53D05 #70H33 #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary: 70H05 #Quantum chaos and dynamical systems #Secondary: 37J10 #Symplectic Geometry (math.SG) #math.SG #msc:37J10 #msc:53D05 #msc:70H05 #msc:70H33
paper · pdf · doi:10.48550/arxiv.1011.5731
26 pages 1 figure
openalex publication_date 2010/11/26 · arxiv created 2011/02/19 · arxiv updated 2011/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a Hamiltonian vector field defined on a symplectic manifold (M,ω), g a nowhere vanishing smooth function defined on an open dense subset M0 of M. We will say that the vector field Y = gX is conformally Hamiltonian. We prove that when X is complete, when Y is Hamiltonian with respect to another symplectic form ω2 defined on M0, and when another technical condition is satisfied, there exists a symplectic diffeomorphism from (M0,ω2) onto an open subset of (M,ω), equivariant with respect to the flows of the vector fields Y on M0 and X on M. This result explains why the diffeomorphism of the phase space of the Kepler problem restricted to the negative (resp. positive) values of the energy function, onto an open subset of the cotangent bundle to a three-dimensional sphere (resp. two-sheeted hyperboloid), discovered by Györgyi (1968) [9], re-discovered by Ligon and Schaaf (1976) [15], whose properties were discussed by Cushman and Duistermaat (1997) [5], is a symplectic diffeomorphism. Infinitesimal symmetries of the Ke- pler problem are discussed, and it is shown that their space is a Lie algebroid with zero anchor map rather than a Lie algebra.