Richard Cushman

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincar´e theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.

In this paper we give two applications of the odd symplectic group to the study of the linear Poincaré maps of a periodic orbits of a Hamiltonian vector field, which cannot be obtained using the standard symplectic theory. First we look at the geodesic flow. We show that the period of the geodesic is a noneigenvalue modulus of the conjugacy class in the odd symplectic group of the linear Poincaré map. Second, we study an example of a family of periodic orbits, which forms a folded Robinson cylinder. The stability of this family uses the fact that the unipotent odd symplectic Poincaré map at the fold has a noneigenvalue modulus.

On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case ($1:-2$) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:-2$ resonance.

In this paper we classify the adjoint orbits of the odd symplectic group over the field of real numbers. We need a noneigenvalue modulus to classify certain orbits.

The aim of this paper is to give a brief description of singular reduction of non-holonomically constrained Hamiltonian systems.

We study the motion of a disk which rolls on a horizontal plane under the influence of gravity, without slipping or loss of energy due to friction. There is a codimension one semi-analytic subset $F$ of the phase space such that the disk falls flat in a finite time, if and only if its initial phase point belongs to $F$. We describe the motion of the disk when it starts at a point $p \notin F$ which is close to a point $f \in F$. It then almost falls flat, after which it rises up again. We prove that during the short time interval that the disk is almost flat, the point of contact races around the rim of the disk from a well defined position at the end of falling to a well defined position at the beginning of rising, where the increase of the angle only depends on the mass distribution of the disk and the radius of the rim. The sign of the increase of the angle depends on the side of $F$ from which $p$ approaches $f$.

We study a nonholonomically constrained Hamiltonian system with a symmetry group which acts properly and freely on a constraint distribution. We show that the reduced dynamics is described by a generalized distributional Hamiltonian system. The general theory is illustrated by the example of Chaplygin's skate.