Canonical perturbation theory via simultaneous approximation.

*(English. Russian original)*Zbl 0795.58042
Russ. Math. Surv. 47, No. 6, 57-133 (1992); translation from Usp. Mat. Nauk 47, No. 6, 59-140 (1992).

In the work, an alternative approach to the Kolmogorov-Arnold-Moser theory is put forward. The approach is based on approximation of an arbitrary trajectory by a closed resonant curve with rational relations between the corresponding frequencies. The author considers the perturbed systems with the Hamiltonian of the form \(H(p,q) = h(p)+\varepsilon f(p,q)\), where \(p\) and \(q\) are \(n\)-component action and angle variables, the full Hamiltonian is an analytical function of its arguments, and (which is a stronger condition than the usual condition of \(h(p)\) being a “steep” function) the unperturbed Hamiltonian \(h(p)\) is assumed to be quasi-convex, i.e., the surface \(h(p)=E\) must be convex in a certain interval of values of the energy \(E\).

The author obtains an improved version of the known Nekhoroshev’s theorem: at the times \(t \leq T (\varepsilon)\) and at \(\varepsilon < \varepsilon_ 0\), the inequality \(\| p(t)-p(0) \| \leq R (\varepsilon)\), where \(R(\varepsilon) = {\mathcal O} (\varepsilon^ b)\) and \(T(\varepsilon)\) is of order of \(\exp (c/ \varepsilon^ a)\). In any case, it is assumed that \(a \leq 1/(2n)\) (recall \(n\) is the number of the degrees of freedom). In the present work, it is proved that, simultaneously, \(a>1/(2n+1)-\eta\) with an arbitrarily small positive \(\eta\).

The author obtains an improved version of the known Nekhoroshev’s theorem: at the times \(t \leq T (\varepsilon)\) and at \(\varepsilon < \varepsilon_ 0\), the inequality \(\| p(t)-p(0) \| \leq R (\varepsilon)\), where \(R(\varepsilon) = {\mathcal O} (\varepsilon^ b)\) and \(T(\varepsilon)\) is of order of \(\exp (c/ \varepsilon^ a)\). In any case, it is assumed that \(a \leq 1/(2n)\) (recall \(n\) is the number of the degrees of freedom). In the present work, it is proved that, simultaneously, \(a>1/(2n+1)-\eta\) with an arbitrarily small positive \(\eta\).

Reviewer: B.A.Malomed (Ramat Aviv)