Larry Bates

Monodromy in the champagne bottle

We show that the Hamiltonian description of a particle moving in a potential field shaped like the punt of a champagne bottle (more properly anS1 symmetric double well) has monodromy, which is a global obstruction to the construction of action-angle variables.

Geometry of nonholonomic constraints

Abstract This paper presents a Hamiltonian treatment of nonholonomically constrained mechanical systems. We assume that the total energy of the systems is a sum of kinetic and potential energies and that the constraints are linear in velocities. We prove a nonholonomic version of Noethers theorem and treat a special case of the nonholonomic reconstruction problem. The entire theory is illustrated by a disc, which rolls on a plane without slipping.

What is a completely integrable nonholonomic dynamical system

Abstract We compare the geometry of a toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. We illustrate their differences using the following examples: the nonholonomic oscillator, Chaplygins skate, Rouths sphere and the rolling oblate ellipsoid of revolution.

Examples of gauge conservation laws in nonholonomic systems

Abstract Some examples of ‘mysterious’ constants of motion in nonholonomic systems are shown to be related to the symmetry group and the constraints in a gauge-like way.

Examples of singular nonholonomic reduction

Abstract We examine the nonholonomic bracket with a view to doing singular nonholonomic reduction. Singular reduction explains not only the seeming paradox of asymptotic stability in Hamiltonian systems but yields a geometrically faithful framework for studying singularity in the tippe top.

Degeneration of Hamiltonian monodromy cycles

Geometric monodromy is an obstruction for the global existence of action variables. The authors study two examples which have nontrivial monodromy and exhibit degeneration phenomena. The first is the classical Kirchhoff case of motion of a rigid body in an infinite ideal fluid. The second is a spherical pendulum subject to an axially symmetric quadratic potential.

On Action-Angle Variables

In this article we give a new proof of the construction of action-angle variables for completely integrable Hamiltonian systems. We also provide a proof of a theorem originally due to EI~RESMANN [5, 17] (which is of interest in its own right) in order to understand the different roles that the differential topology and the symplectic geometry play. Various notes and comments may be found at the end of the proof of the theorem. The chief merit of our approach is the ability to deal with the actions directly as a momentum map. For the sake of definiteness we are working in the category of C ~ functions and mappings. The results are also valid in the analytic and C k categories; for the latter a little bookkeeping on the number of derivatives is needed.

Closed geodesics on the space of stable two-monopoles

We show that the Atiyah-Hitchin metric on the space of stable two-monopoles admits closed geodesics.

Complete integrability beyond Liouville-Arnol'd

Completely integrable Hamiltonian systems with fibres not of cylindrical type are shown to arise naturally from holomorphic functions of two variables. They can have Hamiltonian monodromy of finite order.

You can't get there from here☆

Abstract Not every two points in a compact, connected manifold may be joined by a geodesic of a complete connection.