Abstract: In these talks we describe a bifurcation from a stationary state to a steady whirling motion for an elastic string with fixed ends. We assume that the string is in tension when at rest, it is free to rotate about the axis determined by the stationary state and that no external forces act on the string. Steady whirling at a constant angular velocity is an example of a relative equilibrium. Thus, we use the reduced energy-momentum method to construct an equation whose solutions are possible configurations for steady whirling. One solution to this equation is the stationary state of the string. Using angular velocity as a bifurcation parameter we shall determine the critical angular velocity when non-trivial whirling motion becomes possible. To accomplish this task we use Liapunov-Schmidt reduction to reduce the original infinite dimensional problem to one in finite dimensions. We shall give a general overview of this procedure and discuss how symmetries of the original equation are inherited by the reduced equation. Finally, we use a Taylor expansion to estimate the shape of the string as it moves in steady rotation.

Abstract: Many of the powerful tools of symplectic topology were developed to prove the conjectures of Arnold and Weinstein on the existence of periodic orbits for certain Hamiltonian flows. I will describe joint work with Viktor Ginzburg in which we adapt some of these tools to prove the existence of periodic orbits for other Hamiltonian flows which describe the motion of a charge in a magnetic field. In particular, we prove that for nondegenerate magnetic fields there are periodic orbits on a sequence of low energy levels converging to zero. More generally, we extend the classical Weinstein-Moser theorem by establishing the existence of periodic orbits near a symplectic Bott-nondegenerate extremal submanifold of a Hamiltonian.

Abstract: In the early 80s Duistermaat and Heckman showed that for a Hamiltonian with a quasiperiodic flow on a symplectic manifold the oscillating integral is exactly equal to the leading term of its quasi-classical asymptotics. Since the flow is quasiperiodic it generates a Hamiltonian torus action and this action forces the higher order terms in the asymptotic expansion to vanish. Although the first proof of the Duistermaat-Heckman formula was essentially analytic, the formula brought together analysis and the geometry of group actions on symplectic manifolds. Purely equivariant-topological proofs were soon discovered by Atiyah and Bott and Berline and Vergne. The Duistermaat-Heckman formula and its topological interpretation had a tremendous impact on symplectic geometry, reaching far beyond the scope of the original question.

In these talks we explore the topology and symplectic geometry of the Duistermaat-Heckman formula. In particular, we introduce equivariant cohomology and prove the Atiyah-Bott-Berline-Vergne localization theorem generalizing the Duistermaat-Heckman formula.

Fall 2000: Symplectic Geometry Seminar