I. Aparicio and J. Waldvogel
Applied Mathematics - ETH, Switzerland
Hill's lunar problem describes the planar motion of a massless particle
(the Moon) in the gravitational field of a central body (the Earth) and
an infinitely remote massive perturbing body (the Sun). Alternatively,
it models the close encounter of two small coorbital satellites of a
central body in a rotating frame of reference.
The problem is described by the 2-degree-of-freedom Hamiltonian\\
which may also be written as a polynomial of degree 6 in terms of
Levi-Civita's regularizing variables.
Being the simplest version of the three-body problem, Hill's
lunar problem nevertheless seems to display the full complexity of
nonlinear dynamics, although its non-integrability has not been
rigorously proven to the present day.
In this talk we present new results on the Poincaré map of
Hill's lunar problem with respect to sections with the (x,x dot)-plane.
In particular, fixed points (i.e. periodic orbits), invariant tori of
varios rotation numbers, and chaotic orbits are located and displayed.
In addition, the invariant manifolds of certain fixed points are
investigated.