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August  2015, 35(8): 3827-3855. doi: 10.3934/dcds.2015.35.3827

Continuous averaging proof of the Nekhoroshev theorem

Jinxin Xue 1,

1. 

Department of mathematics, the University of Chicago, Chicago, IL, 60637, United States

Received  August 2013 Revised  December 2014 Published  February 2015

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.
Keywords: Hamiltonian systems., Continuous averaging, Nekhoroshev theorem.
Mathematics Subject Classification: Primary: 37J40, 37J25; Secondary: 34D1.
Citation: Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827
References:
[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.

show all references

References:
[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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