Integrability and non-integrability in Hamiltonian mechanics

Hamiltonian Structure and Integrability

To describe dynamical systems we usually make suitable approximations in the hope of finding valid descriptions of their characteristic quantities. But even after such approximations we mostly cannot write down explicitly how these quantities depend on time, usually such a dependence is much to complicated to be computed explicitly. Therefore we commonly write down dynamical systems in their in...

Noether Symmetries and Integrability in Time-dependent Hamiltonian Mechanics

We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincaré–Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincaré–Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular th...

Symmetry and integrability in Hamiltonian normal forms

Hamiltonian PDEs: deformations, integrability, solutions

We review recent classification results on the theory of systems of nonlinear Hamiltonian partial differential equations with one spatial dimension, including a perturbative approach to the integrability theory of such systems, and discuss universality conjectures describing critical behaviour of solutions to such systems. PACS numbers: 02.30.Ik, 02.30.Jr

Integrability and non-integrability of periodic non-autonomous Lyness recurrences∗

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an +xn+1)/xn, where {an} is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k ∈ {1, 2, 3, 6} the behavior of the sequence {xn} is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases ...