Dynamics, Stability and Symmetric minimization

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The oldest problem in dynamics was uncovered by in the 18th C. by Laplace and Lagrange, namely the question of the stability of the solar system. The equations of Celestial mechanics have been studied for over 300 years, yet the introduction of new methods and techniques of global analysis have created a modern renaissance of results, applications and interconnected research programs. Modern applications include space mission design, molecular dynamics and considerations of dynamics and stability for exoplanet systems discovered recently outside our own solar system. Applications such as these require information about mixtures of stable and hyperbolic motion on invariant sets of the phase space which carry important dynamic properties. The investigation of these invariant sets requires new global geometric techniques. Symmetric minimization in dynamical settings including collision free solutions of the N-body problem is an important new method for studying global periodic orbits and their dynamic neighbourhoods. My research program in the period covered by this proposal will involve projects centred around the theme of characterizing stability or instability of symmetric minimizing periodic orbits. Additional projects using the methods and structures of global symplectic geometry such as using the Maslov index to characterize long period orbits in the neighbourhood of relative equilibria for the spatial restricted three body problem will involve and encourage the participation of HQP. With the help of funding provided by NSERC, my program will continue to provide opportunities for advanced learning and development for undergraduate and graduate students working together with me. My research program continues to be enhanced and deepened by research input and collaboration by current and future graduate students under my supervision. The impact of this fundamental research into the dynamics of the N-body problem will contribute to research groups in Canada and worldwide currently involved in development of fundamental understanding for this important topic.