Martin Kummer

On averaging, reduction, and symmetry in Hamiltonian systems

Abstract The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Henon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order.

On resonant non linearly coupled oscillators with two equal frequencies

AbstractThis paper contains a detailed study of the flow that the classical Hamiltonian

On resonant classical Hamiltonians with two equal frequencies

H = \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} ) + \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} ) + \mathcal{O}_3

Nonintegrability of the classical Zeeman Hamiltonian

induces inR4,

Class of Perturbation Theories of Ordinary Differential Equations

\mathcal{O}_3

Nonlinearly coupled oscillators in quantum mechanics: A normal form approach

representing a convergent power series that begins with a third order term.In particular the existence and stability of periodic orbits is investigated.