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A property of conformally Hamiltonian vector fields; Application to the Kepler problem

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  • Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y=gX$ is \emph{conformally Hamiltonian}. We prove that when $X$ is complete, when $Y$ is Hamiltonian with respect to another symplectic form $\omega_2$ defined on $M^0$, and when another technical condition is satisfied, then there is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$, which maps each orbit to itself and is equivariant with respect to the flows of the vector fields $Y$ on $M^0$ 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) [10], re-discovered by Ligon and Schaaf (1976) [16], is a symplectic diffeomorphism. Cushman and Duistermaat (1997) [5] have shown that the Györgyi-Ligon-Schaaf diffeomorphism is characterized by three very natural properties; here that diffeomorphism is obtained by composition of the diffeomorphism given by our result about conformally Hamiltonian vector fields with a (non-symplectic) diffeomorphism built by a variant of Moser's method [20]. Infinitesimal symmetries of the Kepler problem are discussed, and it is shown that their space is a Lie algebroid with zero anchor map rather than a Lie algebra.
    Mathematics Subject Classification: Primary: 70H05; Secondary: 37J10, 53D05, 70H33.


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