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December  2010, 3(4): 545-578. doi: 10.3934/dcdss.2010.3.545

Properly-degenerate KAM theory (following V. I. Arnold)

Luigi Chierchia 1, and  Gabriella Pinzari 2,

1. 

Dipartimento di Matematica, Università "Roma Tre", Largo S. L. Murialdo 1, 00146 Roma
2. 

Dipartimento di Matematica ed Applicazioni "R. Caccioppoli”, Università di Napoli "Federico II”, Monte Sant’Angelo – Via Cinthia I-80126 Napoli, Italy

Received  April 2009 Revised  May 2010 Published  August 2010

Arnold's "Fundamental Theorem'' on properly-degenerate systems [3, Chapter IV] is revisited and improved with particular attention to the relation between the perturbative parameters and to the measure of the Kolmogorov set. Relations with the planetary many-body problem are shortly discussed.
Keywords: many–body problem, small divisors, invariant tori, degeneracies., KAM theory, Kolmogorov set.
Mathematics Subject Classification: Primary: 70H08, 70K43, 34C27, 34C29, 34D10; Secondary: 70H12, 70K45, 70F15, 70F1.
Citation: Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
References:
[1]

K. Abdullah and A. Albouy, On a strange resonance noticed by M. Herman, Regul. Chaotic Dyn., 6 (2001), 421-432. doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Translated from the Russian by K. Vogtmann and A. Weinstein, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.  Google Scholar

[3]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, (Russian) Uspehi Mat. Nauk, 18 (1963), 91-192; English translation, Russian Math. Surveys, 18 (1963), 85-191.  Google Scholar

[4]

L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. doi: 10.1007/s00205-003-0269-2.  Google Scholar

[5]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134pp.  Google Scholar

[6]

L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem, Ergodic Theory Dynam. Systems, 29 ( 2009), 849-873. doi: 10.1017/S0143385708000503.  Google Scholar

[7]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.  Google Scholar

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.  Google Scholar

[9]

H. Federer, "Geometric Measure Theory,'' Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.  Google Scholar

[10]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), (French)Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582. Revised version (2007) at http://people.math.jussieu.fr/ fejoz/articles.html.  Google Scholar

[11]

H. Hofer, E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,'' Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 1994.  Google Scholar

[12]

U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377-398 (electronic).  Google Scholar

[13]

G. Pinzari, "On the Kolmogorov Set for Many-Body Problems," PhD thesis, Università degli Studi Roma Tre, April 2009, Available at http://ricerca.mat.uniroma3.it/dottorato/tesi.html. Google Scholar

[14]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[15]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions, Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089.  Google Scholar

[16]

H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988), 211-223, Math. Appl., 59, Kluwer Acad. Publ., Dordrecht, 1990.  Google Scholar

[17]

M. B. Sevryuk, The classical KAM theory at the dawn of the twenty-first century, Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3 (2003), 1113-1144, 1201-1202.  Google Scholar

[18]

J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.  Google Scholar

show all references

References:
[1]

K. Abdullah and A. Albouy, On a strange resonance noticed by M. Herman, Regul. Chaotic Dyn., 6 (2001), 421-432. doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,'' Translated from the Russian by K. Vogtmann and A. Weinstein, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.  Google Scholar

[3]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, (Russian) Uspehi Mat. Nauk, 18 (1963), 91-192; English translation, Russian Math. Surveys, 18 (1963), 85-191.  Google Scholar

[4]

L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. doi: 10.1007/s00205-003-0269-2.  Google Scholar

[5]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134pp.  Google Scholar

[6]

L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem, Ergodic Theory Dynam. Systems, 29 ( 2009), 849-873. doi: 10.1017/S0143385708000503.  Google Scholar

[7]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.  Google Scholar

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.  Google Scholar

[9]

H. Federer, "Geometric Measure Theory,'' Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.  Google Scholar

[10]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), (French)Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582. Revised version (2007) at http://people.math.jussieu.fr/ fejoz/articles.html.  Google Scholar

[11]

H. Hofer, E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,'' Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 1994.  Google Scholar

[12]

U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377-398 (electronic).  Google Scholar

[13]

G. Pinzari, "On the Kolmogorov Set for Many-Body Problems," PhD thesis, Università degli Studi Roma Tre, April 2009, Available at http://ricerca.mat.uniroma3.it/dottorato/tesi.html. Google Scholar

[14]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[15]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions, Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089.  Google Scholar

[16]

H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988), 211-223, Math. Appl., 59, Kluwer Acad. Publ., Dordrecht, 1990.  Google Scholar

[17]

M. B. Sevryuk, The classical KAM theory at the dawn of the twenty-first century, Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3 (2003), 1113-1144, 1201-1202.  Google Scholar

[18]

J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.  Google Scholar

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