J.Mikram and F.Zinoun
Département de Mathématiques et Informatique,
Faculté des Sciences de Rabat, Morocco
mikram@fsr.ac.ma
Abstract
Within the scope of studies of qualitative behaviour of nonlinear dynamical systems, we deal with a simple approach to derive normal forms of Hamiltonian vector fields which is especially useful when we have further structures to preserve in the normalization procedure. This algorithm, couched in Lie-Poincaré style, is the Hamiltonian version of Walcher's normal form method1 which we have found simple and elegant. The normalizing transformations inherit the same properties than Walcher's and so they preseve, for example, all structures that can be formulated in terms of graded Lie algebras.
In a second part, we introduce the notion of joint normal forms for Hamiltonian systems using their constants of motion. This allows as to see that the more the Hamiltonian has invariants the more its normal form is simple. Moreover, we show that, under suitable conditions, this ``renormalization'' can be done via a structure preserving transformation. We illustrate our approach by the simple example of the integrable two degree of freedom Hénon-Heiles system.
Finally, we point out that our discussions and statements are completely elementary and explicit so that they can be implemented efficiently in any computer algebra system 2.