Article Contents
Article Contents

# Continuous averaging proof of the Nekhoroshev theorem

• In this paper we develop the continuous averaging method of Treschev to work on the simultaneous Diophantine approximation and apply the result to give a new proof of the Nekhoroshev theorem. We obtain a sharp normal form theorem and explicit estimates of the stability constants appearing in the Nekhoroshev theorem.
Mathematics Subject Classification: Primary: 37J40, 37J25; Secondary: 34D10.

 Citation:

•  [1] A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians, Nonlinearity, 24 (2011), 97-112.doi: 10.1088/0951-7715/24/1/005. [2] A. Córdoba, D. Córdoba and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Annals of Mathematics, 162 (2005), 1377-1389.doi: 10.4007/annals.2005.162.1377. [3] J. Féjoz, M. Guardia, V. Kaloshin and P. Raldan, Kirkwood gaps and diffusion along mean motion resonance in the restricted planar three-body problem, http://arxiv.org/abs/1109.2892. [4] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Math., 167 (2007), 445-453.doi: 10.1007/s00222-006-0020-3. [5] P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Mathematical Surveys, 47 (1992), 57-133.doi: 10.1070/RM1992v047n06ABEH000965. [6] P. Lochak, Simultaneous Diophantine approximation in classical perturbation theory: Why and what for? Progress in nonlinear science., 1 RAS, Inst. Appl. Phys., Nizhnii (Novgorod, (2002), 116-138. [7] P. Lochak and A. I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2 (1992), 495-499.doi: 10.1063/1.165891. [8] P. Lochak, A. I. Neishtadt and L. Niederman, Stability of nearly integrable convex Hamiltonian systems over exponentially long times. Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Diff. Equ. Appl. 12 (1994), 15-34. [9] L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751.doi: 10.1088/0951-7715/9/6/017. [10] N. Nekhorochev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surv., 32 (1977), 5-66, 287. [11] J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Mathematische Zeitschrift, 213 (1993), 187-216.doi: 10.1007/BF03025718. [12] A. Pronin and D. Treschev, Continuous averaging in multi-frequency slow-fast systems, Regular and Chaotic Dynamics, 5 (2000), 157-170.doi: 10.1070/rd2000v005n02ABEH000138. [13] D. Treschev and O. Zubelevich, Introduction to the Perturbation Theory of Hamiltonian Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.doi: 10.1007/978-3-642-03028-4. [14] D. V. Treschev, The continuous averaging method in the problem of separation of fast and slow motions, Regular and Chaotic Dynamics, 2 (1997), 9-20. [15] D. Treschev, Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98.

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