Geometric mechanics, dynamical systems, and symplectic geometry and topology are all very active areas of mathematics having numerous fruitful and deep connections with each other. The underlying theme of the seminar is understanding of these connections.

This quarter we will have talks on a broad variety of topics each concerning some of these areas. The topics will include integration of ODEs on Lie groups and, in the area of mechanics, stability of equilibria of Lagrangian systems. In the field of symplectic geometry, we will discuss, for example, the Duistermaat--Heckman formula and existence of periodic orbits of a charge in a magnetic field. The majority of talks will be introductory and require no prior knowledge of the subject beyond elementary differential and symplectic geometry.

Abstract: We use equivariant techniques to find symplectic embeddings of balls into Grassmannians. This a joint work with Y. Karshon.

Abstract: In this talk we will discuss deformation quantization of line bundles over a symplectic (or Poisson) manifold M. We will show how certain geometric objects (contravariant connections) arising in the semiclassical limit of such deformations can be used to study the classification of star products on M up to Morita equivalence. We will discuss all the necessary background.

Abstract: In symplectic geometry, a classical result of Gromov says that we cannot symplectically embed a ball in Euclidean space inside a cylinder with smaller radius. In contact geometry we cannot expect such a strong non-squeezing property because of the conformal character for contact manifolds. However, we can formulate such theorems for certain contact manifolds with nonzero first Betti number. In this talk, I will prove a non-squeezing theorem for the total space of trivial circle bundle over Euclidean space using relative contact homology theory, and define a contact invariant similar to Gromov width in symplectic geometry.

Abstract: The Poincare recurrence theorem suggests that there should be many periodic orbits on any compact level set of a Hamiltonian flow. Moreover, on the standard symplectic R²ⁿ, Hofer and Zehnder have shown that there are closed orbits on almost every level set of a proper Hamiltonian. With these results in mind, it seems reasonable to guess that there are periodic orbits on every compact level set of a Hamiltonian flow on R²ⁿ. Indeed, this assertion even has a name, the Hamiltonian Seifert conjecture (after a similar conjecture by Seifert concerning flows on the three sphere).

Although it remains open for n=2 (i.e., R⁴), counterexamples to this conjecture for n>2 were constructed, independently and in different ways, by Viktor Ginzburg and Michel Herman. In particular, they both describe the construction of smooth functions which have a level set that is C⁰-close to the unit sphere but contains no periodic orbits. In this talk, we will describe a new and comparatively simple procedure for building such functions. This is a modification of the work of Ginzburg.

Abstract: The system of three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic and cubic Hamiltonians are shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associtated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three waves can be controlled. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type-I and -II second harmonic generation, and for parametric three-wave interactions.

References:

Alber, M.S., G.G. Luther, J.E. Marsden and J.M. Robbins [1998], Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction, Physica D 123, 271-290

Alber, M.S., G.G. Luther, J.E. Marsden and J.W. Robbins [1999], Geometry and Control of Three-Wave Interactions, Fields Inst. Commun. 24, 55-80.

Alber, M.S., G.G. Luther, J.E. Marsden and J.W. Robbins [2000], Geometry and control of x⁽²⁾ processes and the generalized Poincar\'e sphere, J. Opt. Soc. Amer.B. 17 , 6, 932-941.

Abstract: The motion of a rigid body pinned at a point can be described by a time-parametrized sequence of orthogonal transformations. The Lagrange top is a pinned rigid body in gravitational force field, with a mass distribution that is symmetric about an axis. As everyone knows, if the top is spinning fast enough, then it "goes to sleep", i.e., it spins almost exactly upright. As it slows down, this vertical spinning becomes unstable. At some point, the top starts to tilt and precess. This is an example of a bifurcation from an isotropic relative equilibrium.

A pseudo-rigid body is a generalization of the rigid body that allows deformation by any nondegenerate linear transformation. The type of material the body is made of determines the way the body resists deformation. In the hyperelastic case, this resistive force is determined by a scalar function called the internal potential energy.

A pseudo-rigid body is a generalization of the rigid body that allows deformation by any nondegenerate linear transformation. The type of material the body is made of determines the way the body resists deformation. In the hyperelastic case, this resistive force is determined by a scalar function called the internal potential energy.

A pseudo-Lagrange top is like a Lagrange top, except that it is made of a pseudo-rigid material. (The reference body is assumed to be axisymmetric, but can deform to asymmetric shapes.) This talk is about the bifurcation of a sleeping pseudo-Lagrange top. A steadily processing top can be characterized as a critical point of the augmented potential of the system. We apply Liapunov-Schmidt Reduction and the Equivariant Branching Lemma to the differential of the augmented potential and derive a surprisingly simple test for the bifurcation of the sleeping pseudo-Lagrange top. The computations involved are greatly simplified by the existence of the reflectional symmetry of the system. This use of discrete symmetry is somewhat "new" to the study of relative equilibria in Lagrangian/Hamiltonian systems with symmetry. The techniques used here can be applied to a wide variety of other symmetric Lagrangian systems.

Abstract: The Kepler problem is the ODE which governs the motion of two planets under their mutual attraction, or, one planet under the influence of an infinitely heavy star. Its orbits are well-known: they are conic sections with one focus at the star. What is less well-known is that the Kepler equation can be `regularized' so as to become geodesic flow on the sphere. The collision orbits of Kepler become geodesics through the North pole. We will describe this 30 years old result of Moser, and time permitting, related results of Levi-Civita, Steiffel, and Milnor.