Dynamics and Symmetry: an Excursion Beyond the Hill Boundary
Prof. Daniel Offin
Queen's University, Kingston, Canada
Abstract: In the study of conservative Hamiltonian systems, stability of peri- odic orbits play an essential role in understanding the global dynamical process from the equations of motion. Two important examples of this important aspect of investigation are exhibited in the Henon Hieiles Hamiltonian for energies less than 1/6, as well as the gravitational sys- tem described by the three body problem of Celestial Mechanics, which inclludes the famous gure eight solution, discovered by Chenciner- Montgomery, Moore, when angular momentum is zero and energy is negative. Important aspects are the determination of periodic orbits and their stability type which are far from integrable or equilibrium states. The gure eight exempli es this aspect. The variational calculus is an im- portant and powerful tool in this regard. Stability theory for conserva- tive systems precludes asymptotic stability since volume is preserved by the ow of such systems. For conservative systems there can be a mixture of stable and unstable motion. The extremes are hyper- bolic behaviour (completely unstable) and elliptic stability where the linearized equations essentially split as multiple copies of harmonic os- cillators. Both of these are important and have aspects which are useful for di erent lines of inquiry. Hyperbolic behaviour can often be linked to chaotic dynamics (such as in the Henon-Heieles Hamiltonian and the three body problem) whereas elliptic stability has important appli- cations in engineering design. In this study we will investigate several instances of hyperbolic behavioiur which has important implications for chaotic dynamics in applications. Newton's equations for kinetic plus potential Hamiltonian are re- formulated into geodesic equations for xed energy E via the Jacobi- Maupertuis [JM] metric. This metric is formed with a conformal factor times the Riemannian metric of the kinetic energy. The conformal fac- tor vanishes on the zero velocity surface, where the potential function takes the value E. This equipotential surface is sometimes called the Hill boundary when investigating the properties of geodesics in the JM metric. The JM metric degenerates on the Hill boundary and this causes diculties with analysis of geodesics which intersect the bound- ary. Montgomery [] has made a masterful exposition of the Lagrangian singularity occuring at and near the boundary. This is critical for un- derstanding how the JM geodesics evolve near a point of intersection of a geodesic with the Hill boundary and plays a signi cant role in the current study. Earlier works by Seifert [] used the JM metric to estab- lish the existence of periodic orbits of Newton's equations lying within compact potential bowls by establishing the existence of JM geodesics which intersect the boundary in two distinct points, now knows as brake orbits. Our goal is to present an index theory for such class of closed orbits, and applications to the study of their stability status.
Brief Biography of the Speaker: To be anancioud soon