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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 & Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827
References:
[1]

Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[2]

Annals of Mathematics, 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.  Google Scholar

[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,, , ().   Google Scholar

[4]

Inventiones Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[5]

Russian Mathematical Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[6]

Progress in nonlinear science., 1 RAS, Inst. Appl. Phys., Nizhnii (Novgorod, (2002), 116-138.  Google Scholar

[7]

Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[8]

Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Diff. Equ. Appl. 12 (1994), 15-34.  Google Scholar

[9]

Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[10]

Russ. Math. Surv., 32 (1977), 5-66, 287.  Google Scholar

[11]

Mathematische Zeitschrift, 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[12]

Regular and Chaotic Dynamics, 5 (2000), 157-170. doi: 10.1070/rd2000v005n02ABEH000138.  Google Scholar

[13]

Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03028-4.  Google Scholar

[14]

Regular and Chaotic Dynamics, 2 (1997), 9-20.  Google Scholar

[15]

Russian J. Math. Phys., 5 (1997), 63-98.  Google Scholar

show all references

References:
[1]

Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[2]

Annals of Mathematics, 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.  Google Scholar

[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,, , ().   Google Scholar

[4]

Inventiones Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[5]

Russian Mathematical Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[6]

Progress in nonlinear science., 1 RAS, Inst. Appl. Phys., Nizhnii (Novgorod, (2002), 116-138.  Google Scholar

[7]

Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.  Google Scholar

[8]

Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Diff. Equ. Appl. 12 (1994), 15-34.  Google Scholar

[9]

Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[10]

Russ. Math. Surv., 32 (1977), 5-66, 287.  Google Scholar

[11]

Mathematische Zeitschrift, 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[12]

Regular and Chaotic Dynamics, 5 (2000), 157-170. doi: 10.1070/rd2000v005n02ABEH000138.  Google Scholar

[13]

Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-03028-4.  Google Scholar

[14]

Regular and Chaotic Dynamics, 2 (1997), 9-20.  Google Scholar

[15]

Russian J. Math. Phys., 5 (1997), 63-98.  Google Scholar

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