Richard Cushman

The quantum mechanical spherical pendulum

On decrit le comportement asymptotique du spectre du pendule quantique spherique quand la constante de Planck tend vers zero

Monodromy in the hydrogen atom in crossed fields

Abstract We show that the hydrogen atom in orthogonal electric and magnetic fields has a special property of certain integrable classical Hamiltonian systems known as monodromy. The strength of the fields is assumed to be small enough to validate the use of a truncated normal form H snf which is obtained from a two step normalization of the original system. We consider the level sets of H snf on the second reduced phase space. For an open set of field parameters we show that there is a special dynamically invariant set which is a “doubly pinched 2-torus”. This implies that the integrable Hamiltonian H snf has monodromy. Manifestation of monodromy in quantum mechanics is also discussed.

Conjugacy classes in linear groups

Abstract Let G belong to one of the three families of complex classical linear groups or to one of the seven families of corresponding real forms. Let L denote its Lie algebra. We give a simple and effective method for finding all conjugacy classes of G and all orbits of G in L . We also describe the splitting of classes and orbits when G is replaced by a normal subgroup. We discuss the situation for other fields.

Reduction of the semisimple 1:1 resonance

The method of “averaging” is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated “reduced” Hamiltonian on a “reduced space”. This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case. The reduced space will be a 2-sphere in R3, and the reduced differential equations will be Eulers equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on “degeneracies” due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.

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.

Reduction, Brouwer's Hamiltonian, and the critical inclination

The reduction process is applied twice to the Brouwer Hamiltonian of artificial satellite theory to obain a reduced HamiltonianMH, Lon a reduced phase spacePH, Lwhich is diffeomorphic to a twosphere. To first order in the oblateness ε, the reduced Hamiltonian has two nondegenerate critical points of index 2 and a nondegenerate critical circleC of index 0 at the critical inclinations2-4/5. To second order in ε,MH,Lis a Morse function onPH,L,with three pairs of critical points of index 0.1, and 2, respectively.

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.

Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues

This note gives a concise algorithm for computing a normal form for a real linear Hamiltonian differential equatin which has purely imaginary eigenvalues. This algorithm is then applied to the differential equation which comes from the quadratic terms of the Hamiltonian of the restricted three body problem at a Lagrange equilateral triangle equilibrium point.

Geometry of KAM tori for nearly integrable Hamiltonian systems

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

Geometry of nonholonomically constrained systems

Nonholonomically Constrained Motions Group Actions and Orbit Spaces Symmetry and Reduction Reconstruction, Relative Equilibria and Periodic Orbits Caratheodorys Sleigh Convex Rolling Rigid Body The Rolling Disk.