A geometric setting for Hamiltonian perturbation theory (Q2750966)

The book contains a survey of some methods of Hamiltonian perturbation theory with particular emphasis on the geometric formulation of the problem. As stated by the author, the main goal of the book is to revisit the perturbation theory of non-commutatively integrable systems from the point of view of non-abelian symmetry groups. Non-commutativity arises in the case when the system possesses a set of first integrals that are not in involution. Under such conditions, the phenomenon of degeneration of frequencies occurs and may be tackled by introducing a partial set of action-angle coordinates, in view of a work of \textit{N. N. Nekhoroshev} [Russ. Math. Surv. 32, No.~6, 1-65 (1977; Zbl 0389.70028)]. The author of the present book takes a different approach that exploits the symmetries exhibited by the system. NEWLINENEWLINENEWLINEThe theorem of Nekhoroshev on exponential stability refers to small perturbations of an integrable system. In simple terms, it states that if the size of the perturbation, \(\epsilon\) say, is small enough, then the actions \(p\) of the system satisfy \(\bigl|p(t)-p(0)\bigr|\leq r_0\epsilon^b\) for \(|t|\leq t_0 \exp(c\epsilon^a)\), with positive constants \(a, b, c, r_0\) and \(t_0\). The extension of the theorem to the non-commutative case presents some difficulties due to a motion of fast drift that may take orbits out of the domain where the local action-angle coordinates are defined. NEWLINENEWLINENEWLINEThe author proposes to avoid the latter difficulty by introducing a coordinate system that is intrinsic to the geometry of the system. According to the author, in this context the Nekhoroshev's estimates turn out to have a natural interpretation in terms of momentum maps and co-adjoint orbits. NEWLINENEWLINENEWLINEThe book includes a complete proof of the theorem. The scheme of proof is different from the original one proposed by Nekhoroshev. The proof presented here is based on the work of \textit{P. Lochak} [see, e.g., J. Nonlinearity 6, No. 6, 885-904 (1993; Zbl 0794.70006)]. An application of the theory to the classical problem of the Euler-Poinsot rigid body is also included, together with other examples.