Generalized Normal Forms by Relegation

Andre Deprit
Departamento de Fisica Teorica, Universidad de Zaragoza
Jesus Palacian Subiela
Universidad Publica de Navarra, Pamplona
Etienne Deprit
Computer Science Division, University of California at Berkeley
J.-F. San Juan
Universidad de la Rioja, Logrono


Given a perturbed Hamiltonian system, normalization by Lie series converts the main part of the Hamiltonian into an integral of the transformed system. A generalization of normalization -- the relegation algorithm -- does the same for an arbitrary function G of the state variables. If the Lie derivative defined by G is semi-simple, a double recursion produces the generator of the relegating Lie transformation. We demonstrate the usefuleness of relegation on a simple example consisting of a coupled oscillator and diffusor. Classical normalization of this system yields the possible partitions of total energy between oscillator and diffusor. Relegation, on the other hand, isolates the effect of the perturbation on the fundamental frequency of the oscillator for a given energy of the diffusor. In addition, relegation produces a straightforward solution on the boundary of the parameter domain where Birkhoff normalization fails.


IMACS ACA'98 Electronic Proceedings