In this article, we present a new approach of Nekhoroshev's theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

Using a scheme given by Lochak, we derive a result of stability over exponentially long times with respect to the inverse of the distance to an elliptic equilibrium point which has a definite torsion. At the price of this assumption, our study is valid without arithmetical properties of the linearized system while the previous theorems of this kind rely on a Diophantine condition on the linear spectrum. Actually, under the latter condition and a definite torsion, a result of stability over superexponentially long times can be proved. Finally, the same kind of theorems are also valid for an elliptic lower-dimensional invariant torus.

In the 1970s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system with a perturbation of size $\varepsilon$, the actions linked to the unperturbed Hamiltonian vary only by the order of $\varepsilon^b$ over a time of the order of $\exp (C\varepsilon^{-a})$ for some positive constants a, b and C, provided that the unperturbed Hamiltonian meets some generic transversality condition known as steepness. Among steep systems, convex or quasiconvex systems are easier to describe since the use of energy conservation allows the proof of exponential estimates of stability to be shortened. In this case, Lochak–Neishtadt and Poschel have independently obtained the stability exponents a = b = 1/2n for systems of n degrees of freedom—especially the time exponent (a) is expected to be optimal (see P. Lochak, J.-P. Marco and D. Sauzin. Preprint. Institut de Máthematique de Jussieu, 1999; J.-P. Marco and D. Sauzin. Preprint. Publ. Math. Inst. Hautes Etudes Science, 2001). Moreover, Lochak's study relies on simultaneous Diophantine approximation which gives a very transparent proof.However, the proof in the steep case has rarely been studied since Nekhorochev's original work despite various physical examples where the model Hamiltonian is only steep. Here, we combine the original scheme with a simultaneous Diophantine approximation as in Lochak's proof. This yields significant simplifications with respect to Nekhorochev's reasoning: it also allows the exponents $a=b=(2n p_1\dotsb p_{n-1})^{-1}$ where $(p_1\dotsb p_{n-1})$ are the steepness indices of the considered Hamiltonian to be obtained. In the quasiconvex case, the steepness indices are all equal to one and we find the same exponents 1/2n as Lochak–Neishtadt and Poschel, whose results are thus generalized to the steep case.