## 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 $$\eta$$.

### MSC:

 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 34E10 Perturbations, asymptotics of solutions to ordinary differential equations

### Keywords:

KAM theorem; invariant tori; Arnold’s diffusion
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