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

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]