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

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

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.
Citation: Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete and 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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).

[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.

[14]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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).

[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.

[14]

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

[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.

[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.

[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.

[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.

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