Séminaire de géométrie hamiltonienne (Paris 6) – Janvier 2017
Campus Jussieu, vendredi 10:30 (salle 15-25.502)
Organisateurs : M. Chaperon, M. Irigoyen, L. Lazzarini, J.-P. Marco, J.-B. Caillau
Vendredi 13 et 20 décembre à 10H30 : Andrei A. Agrachev (SISSA) – Symplectic Geometry of Constrained Optimization and Optimal Control
Abstract. Given a conditional optimization problem, its Lagrange multipliers are points of the cotangent bundles to the space of conditions. Cotangent bundle is endowed with the standard symplectic form, and symplectic geometry of the set of Lagrange multipliers allows to efficiently characterize optimal solutions even for very degenerate problems with complicated constraints. A basic instrument is the Maslov index of a triple of Lagrange subspaces. Being approprialy re-arranged, this classical geometric invariant also becomes a powerful analytic tool. As expected, the symplectic approach clarifies and unifies many known facts and opens new horizons.
Vendredi 27 janvier à 10h30 : Guowey Yu (Toronto) – Simple choreographies in the planar Newtonian N-body problem with equal masses
Abstract. In 2000, using variational method, A. Chenciner and R. Montgomery proved the famous “Figure-Eight” solution of the three body problem with equal masses. The remarkable feature of this solution is that all three masses travel on a single loop in the shape of figure 8. For arbitrary N body problem with equal masses, many solutions with the particular feature that all masses travel on a single loop with different shapes were found numerically by C. Simo afterwards and he named them “simple choreographies”. Among them, there is a special family called “linear chain”, where the single loop looks like a sequence of consecutive bubbles. In this talk, we give a proof of the existence of this family.
