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

TL;DR

This paper studies conformally Hamiltonian vector fields, proving a symplectic equivalence under certain conditions, and applies this to explain the symplectic structure of the Kepler problem's phase space.

Contribution

It establishes conditions under which conformally Hamiltonian vector fields are symplectically equivalent and applies this to clarify the geometric structure of the Kepler problem.

Findings

Proves existence of symplectic diffeomorphism under specific conditions.

Explains the symplectic nature of the Kepler problem's phase space.

Shows infinitesimal symmetries form a Lie algebroid, not a Lie algebra.

Abstract

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 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, there exists a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$, 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 Gy\"orgyi (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.