# American Institute of Mathematical Sciences

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.

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##### 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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