Asymptotic Hamiltonian dynamics: the Toda lattice, the three-wave interaction and the non-holonomic Chaplygin sleigh

In this paper we discuss asymptotic stability in energy-preserving systems which have an almost Poisson structure. In particular we consider a class of Poisson systems which includes the Toda lattice. In standard Hamiltonian systems one of course does not expect asymptotic stability. The key here is the structure of the phase space of the Poisson or almost Poisson systems and the nature of the equilibria. Our systems are in some sense generalizations of the integrable Toda lattice system but are not integrable in general. As a particular example we point out an interesting connection between three mechanical two degree-of-freedom systems that exhibit asymptotic stability. Two of them are classical Hamiltonian systems while the third is a non-holonomic system. Non-holonomic systems are generally energy-preserving but not Hamiltonian, but the system analyzed here turns out to have a phase space which is the union of Hamiltonian ones. We also discuss various higher-dimensional examples.