Cometary Escape in the Restricted Circular Planar Three Body Problem

Abstract

The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. This principle is utilized along with Aubry-Mather theory to construct regions of instability for a certain three body problem, given by a Hamiltonian system of two degrees of freedom. In principle, these methods can be applied to construct instability regions for a variety of Hamiltonian systems with $2$ degrees of freedom. The Hamiltonian model considered in this thesis describes the dynamics of a Sun-Jupiter-Comet system and under some simplifying assumptions, the existence of instabilities for the orbit of the comet is shown. In particular it is shown that a comet which starts close to an orbit in the shape of an ellipse of eccentricity $e=0.66$ can increase in eccentricity to beyond $e=1$. Furthermore, there exist ejection orbits for the comet. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity. Several new theoretical tools are introduced in this thesis as well. The most notable is a checkable sufficient condition to verify that an exact area preserving map is an exact area preserving twist map in a certain coordinate system. This coordinate system is constructed by ``spreading the cumulative twist'' which arises from the long term dynamics of system. Many of the results of the thesis are `computer assisted' and utilize recent advances in rigorous numerical integration. It is through the application of these advances in computing that it has become possible to state deep results for realistic solar systems. This has been the dream of many since humans first observed the stars so long ago.