# 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 & 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.  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, (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.   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.  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).   Google Scholar [6] L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem,, Ergodic Theory Dynam. Systems, 29 (2009), 849.  doi: 10.1017/S0143385708000503.  Google Scholar [7] A. Deprit, Elimination of the nodes in problems of $n$ bodies,, Celestial Mech., 30 (1983), 181.  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, (1992).   Google Scholar [9] H. Federer, "Geometric Measure Theory,'', Die Grundlehren der mathematischen Wissenschaften, (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.   Google Scholar [11] H. Hofer, E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,'', Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, (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.   Google Scholar [13] G. Pinzari, "On the Kolmogorov Set for Many-Body Problems,", PhD thesis, (2009).   Google Scholar [14] J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems,, Math. Z., 213 (1993), 187.  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.  doi: 10.1007/BF00692089.  Google Scholar [16] H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems,, Stochastics, (1988), 211.   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, 3 (2003), 1113.   Google Scholar [18] J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems,, Amer. J. Math., 58 (1936), 141.  doi: 10.2307/2371062.  Google Scholar

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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.  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, (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.   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.  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).   Google Scholar [6] L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem,, Ergodic Theory Dynam. Systems, 29 (2009), 849.  doi: 10.1017/S0143385708000503.  Google Scholar [7] A. Deprit, Elimination of the nodes in problems of $n$ bodies,, Celestial Mech., 30 (1983), 181.  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, (1992).   Google Scholar [9] H. Federer, "Geometric Measure Theory,'', Die Grundlehren der mathematischen Wissenschaften, (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.   Google Scholar [11] H. Hofer, E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,'', Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, (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.   Google Scholar [13] G. Pinzari, "On the Kolmogorov Set for Many-Body Problems,", PhD thesis, (2009).   Google Scholar [14] J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems,, Math. Z., 213 (1993), 187.  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.  doi: 10.1007/BF00692089.  Google Scholar [16] H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems,, Stochastics, (1988), 211.   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, 3 (2003), 1113.   Google Scholar [18] J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems,, Amer. J. Math., 58 (1936), 141.  doi: 10.2307/2371062.  Google Scholar
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