Types of integrability on a generalizations of Gordon's theorem (Q2841122)

This profound and substantial paper by the late N.N. Nekhoroshev (1946--2008) deals with (generally, non integrable) autonomous Hamiltonian systems on a symplectic manifold \((M,\omega)\); these are defined by a Hamiltonian function \(H : M \to R\). In some cases these are integrable in the standard or even in the ``non regular'' sense [the author et al., C. R., Math., Acad. Sci. Paris 335, No. 11, 985--988 (2002; Zbl 1017.81009)], but most frequently this is not the case. In a number of situations, the system is integrable in some (dynamically invariant) submanifold \(M_0 \subset M\), but not on the whole \(M\), and this situation is the main theme of this comprehensive paper.NEWLINENEWLINENEWLINENEWLINEFirst of all the standard (toric) integrability on \(M_0\) is considered; in this case there are \(k\)-dimensional invariant tori which provide a locally trivial fibration of \(M_0\). The problem tackled here is that of \textit{finding} the integrability submanifolds, and also studying the \textit{quality of integrability} on them. In the ``highest quality'' case, a generalization of Gordon theorem [\textit{W. B. Gordon}, J. Math. Mech. 19, 111--114 (1969; Zbl 0179.42004)] is obtained. Note that periodic solutions are a special case of integrability submanifolds; this point of view is indeed already present in early work by the author [Funct. Anal. Appl. 28, No. 2, 1 (1994; Zbl 0847.58035); translation from Funkts. Anal. Prilozh. 28, No. 2, 67--69 (1994)] and has been a constant theme in his scientific activity.NEWLINENEWLINENEWLINENEWLINEIn the case where \(M_0\) is also a symplectic manifold, the restriction of the system to \(M_0\) is also Hamiltonian, say with Hamiltonian \(H_0\), and the problem reduces to that of complete integrability of \(H_0\) on the symplectic manifold \((M_0 , \omega_0)\) -- where of course \(\omega_0\) is the restriction of \(\omega\) to \(M_0\). But different situations are also possible (e.g., \(M_0\) could be of odd dimension, thus surely not symplectic). In this case the dynamical system on \(M_0\) is not Hamiltonian, but still it originates from a Hamiltonian system on the larger manifold \(M\). Thus two problems should be considered: (a) how can one take into account the Hamiltonian nature of the ``parent'' system in this case? (b) Under what conditions the generalization of Gordon theorem remain valid?NEWLINENEWLINENEWLINENEWLINEIn order to tackle these questions, one should consider the different types of integrability that a Hamiltonian system can have on a manifold; and in order to study this problem, one is led to consider Liouville integrability (or, as we would now say, Poincaré-Liouville-Lyapounov-Arnold-Nekhoroshev integrability) for general dynamical systems, that is systems which are not necessarily Hamiltonian.NEWLINENEWLINENEWLINENEWLINEClarifying this matter (which is done in Section 2, after recalling some basic facts in Section 1), and classifying the different types of integrability -- in particular, distinguishing between global and local integrability on a submanifold -- opens the way to a discussion for Hamiltonian systems and their restriction to an integrability submanifold (Section 3). A special attention is given (Section 4) to systems which are Hamiltonian and globally integrable on a submanifold \(M_0\); and an extension of Gordon theorem for such systems is also provided (Section 5).NEWLINENEWLINENEWLINENEWLINEIn particular, the following two results (reported here in a sketchy form and with a standard notation) are given as Theorems 5.5.1.A and 5.5.1.B.NEWLINENEWLINENEWLINENEWLINETheorem A considers the fibration of an arbitrary submanifold \(M_0 \subseteq M\) into isotropic tori defined by vector fields \(\{ J d Z_1 ,\dots , J d Z_k \}\), which are not necessarily pairwise commutative on \(M_0\) (thus one has a pseudo-Hamiltonian toric fibration). It is proved that the action functions \(\{ I_1 ,\dots, I_k \}\) are locally constant on the intersection of the common level surfaces \(Z^{-1}\) of the \(Z_i\) with \(M_0\).NEWLINENEWLINENEWLINENEWLINETheorem B asserts that if a system is Hamiltonian-torically \(Z\)-integrable on \(M_0\) (see above), then the frequencies \(\omega = \{ \omega_1 , \dots, \omega_k \}\) of the conditionally periodic motions on the invariant tori (which are the fibers of the corresponding fibration of \(M_0\)) depend locally only on the functions \(Z_i\) defining the fibration, i.e. \(\omega = \omega (Z)\); thus they depend at most on as many parameters as is the dimension of the corresponding invariant tori. It is also shown that under these hypotheses (stronger than those of theorem A), \(H\), \(I\), and \(\omega\) satisfy the same relations as in the symplectic case, i.e. \(\omega = \omega (I) = (\partial H / \partial I)\).NEWLINENEWLINENEWLINENEWLINEAccording to the author, these are the main results of the paper (this reviewer instead find it difficult to choose among the many results and the beautiful constructions provided here); they were announced previously [the author, Regul. Chaotic Dyn. 7, No. 3, 239--247 (2002; Zbl 1019.37035)], but the proof was only outlined.NEWLINENEWLINENEWLINENEWLINEA special Section (Section 6) is devoted to provide detailed proofs of several statement given before, relegated here in order not to interrupt the flow of discussion with detailed and necessarily long proofs.NEWLINENEWLINENEWLINENEWLINEFinally, Section 7 is devoted to the problem of finding the integrability submanifold, i.e. to making the previous theory constructive and readily applicable in the study of concrete systems.NEWLINENEWLINENEWLINENEWLINEThe paper contains a wealth of never trivial remarks opening connections with many topics in mechanics.