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12 Perturbation Theory in Algebraic Setting

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rencontre du non-lin´eaire 2019 1

Perturbation Theory in Algebraic Setting

Lorenzo Valvo & Michel Vittot

Centre de Physique Theorique, Campus de Luminy, 13288 Marseille [email protected]

We propose a perturbation algorithm which works on any Lie algebra V. We consider a dynamical system on V, which preserves a subalgebra B of V. In general when we add a perturbing term to the original dynamical system, the subalgebra Bwon’t be preserved anymore. We show that under suitable hypothesis, there exists a new dynamical system, conjugated to the perturbed one, which preservesBup to terms quadratic in the perturbation. By this formula, classical perturbation theory can be extended to noncanonical Poisson system : for instance, the Euler-Poinsot equations for a rigid body, ideal MHD, the Maxwell-Vlasov and Vlasov-Poisson systems. In particular, we show that, for a time dependent top, it is possible to iterate the formula and to get a KAM theorem : for a large set of initial data, the algebra B, like the tori of classical mechanics, get deformed into a new algebra preserved by the perturbed flow.

c Non Lin´eaire Publications, Avenue de l’Universit´e, BP 12, 76801 Saint- ´Etienne du Rouvray cedex

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