# Mechanics, Symplectic Geometry, and Dynamical Systems

Symplectic geometry is the geometry underlying classical mechanics. It is also important to quantum mechanics and low-dimensional topology and is an active area of research. In the department, three of our faculty represent symplectic geometry and its links to mechanics and dynamics.

**Viktor Ginzburg** is internationally known for achievements in the Hamiltonian Seifert conjecture and for work in Poisson topology. Hamiltonian systems are the natural dynamical systems for a symplectic geometry. Each such system has a "Hamiltonian", or energy, which is constant along solutions to the system. The Hamiltonian Seifert conjecture proposes that if the Hamiltonian has a sphere as one of its level sets, then on that sphere there is a periodic orbit. Ginzburg and his graduate students have constructed lowest dimensional counterexamples to this conjecture, i.e. examples showing that the conjecture is wrong as stated. In contrast, fundamental theorems in symplectic topology assert that Hamiltonian systems should have a certain number of periodic orbits that number dictated by topology. So the work of Ginzburg and his students has provided strong constraints and deepened understanding within symplectic topology.

**Debra Lewis** is an expert in relative equilibria for Hamiltonian systems and their stability and bifurcations relative equilibria. A relative equilibrium for a dynamical system is a solution that is an equilibrium modulo a symmetry group action. Her work is an interplay between group theory, symplectic geometry, and uses a good deal of symbolic manipulation.

**Richard Montgomery** is known for his work on the N-body problem and sub-Riemannian geometry. He and Alain Chenciner (Paris 7 and the Bureau des Longitudes, Paris) rediscovered, and proved the existence of a figure eight shaped orbit for the Newtonian three-body problem. Their result and the introduction of variational and group-theoretic methods to celestial mechanics that they pioneered generated much recent work in mathematical celestial mechanics. He has one basic result in the field ("geodesics which do not satisfy the geodesic equations") and has written a well-received book on it. This work on sub-Riemannian geometry grew out of applications to nonlinear control theory, and physical chemistry (e.g., NMR, Berry's phase).