# 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

show all references

##### 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
 [1] E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401 [2] Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55 [3] Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613 [4] Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941 [5] Rodica Toader. Scattering in domains with many small obstacles. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 321-338. doi: 10.3934/dcds.1998.4.321 [6] Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57 [7] Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64 [8] Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683 [9] Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162 [10] Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433 [11] Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377 [12] Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371 [13] Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633 [14] Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147 [15] Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 [16] Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 [17] Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 [18] Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092 [19] Tim Gutjahr, Karsten Keller. Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4207-4224. doi: 10.3934/dcds.2019170 [20] C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457

2018 Impact Factor: 0.545