Stability and bifurcations of relative equilibiria in simple mechanical systems

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Many mechanical systems can be modeled as Hamiltonian systems, in which the dynamics is determined by a function (the Hamiltonian) and the geometry of the phase space (Poisson or symplectic). Understanding the implications of the geometric structure is central to the analysis of the motion, especially when symmetries are present. An important subclass of mechanical systems is that of simple mechanical systems, where the motion is determined by a kinetic term and a potential term. The presence of continuous symmetries in a mechanical system has profound implications for the dynamics. In such systems, relative equilibria are the simplest non-trivial solutions. They are solutions that move with constant velocity along the symmetry group orbit solution. For instance, a relative equilibrium of the Newtonian 3-body problem is a motion in which each body moves uniformly in a circular orbit around the centre of mass while the shape formed by the 3 mass points is unchanged. Similarly to equilibria for non-symmetric systems, relative equilibria are organizing centres of more complicated dynamics and they can be used as base points for perturbation theory. My research proposal has two main goals. The first is to add to the constructive theory needed to study the dynamics near relative equilibria for symmetric Hamiltonian systems, and in particular, simple mechanical systems. ``Constructive theory" here means a theory which can be applied to concrete examples. Special attention is required near singular points of the symmetry, where a slight perturbation of the system parameters might induce radical changes in behavior. For instance, this situation appears in symmetry breaking phenomena, where the symmetry of the relative equilibrium changes.  Many traditional methods may have to be modified. The second main goal of this proposal is to develop applications with relevance to molecular physics and chemistry (bifurcations from relative equilibria for molecules), celestial mechanics (nonlinear stability of relative equilibria in the N-body problem, N>2) and astrodynamics (dynamics of asteroid pairs).