December  2010, 3(4): 533-544. doi: 10.3934/dcdss.2010.3.533

Some KAM applications to Celestial Mechanics

1. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma

Received  April 2009 Revised  May 2010 Published  August 2010

The existence of invariant tori in Celestial Mechanics has been widely investigated through implementations of the Kolmogorov-Arnold-Moser (KAM) theory. We provide an introduction to some results on the existence of maximal and low-dimensional, rotational and librational tori for models of Celestial Mechanics: from the spin--orbit problem to the three-body and planetary models. We also briefly review a result on dissipative invariant attractors for the spin-orbit problem, whose existence is proven through a dissipative KAM theorem.
Citation: Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Uspehi Mat. Nauk, 18 (1963), 91.   Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopaedia of Mathematical Sciences {\bf 3}, 3 (2006).   Google Scholar

[3]

A. Berretti, A. Celletti, L. Chierchia and C. Falcolini, Natural boundaries for area-preserving twist maps,, J. Stat. Phys., 66 (1992), 1613.  doi: 10.1007/BF01054437.  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]

L. Biasco, L. Chierchia and E. Valdinoci, $N$-dimensional elliptic invariant tori for the planar $(N+1)$-body problem,, SIAM Journal on Mathematical Analysis, 37 (2006), 1560.  doi: 10.1137/S0036141004443646.  Google Scholar

[6]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order amidst Chaos,", Lecture Notes in Mathematics {\bf 1645}, 1645 (1645).   Google Scholar

[7]

H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.  doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[8]

A. Celletti, Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance (Part I),, J. of Applied Math. and Physics (ZAMP), 41 (1990), 174.   Google Scholar

[9]

A. Celletti, Construction of librational invariant tori in the spin-orbit problem,, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61.  doi: 10.1007/BF00942847.  Google Scholar

[10]

A. Celletti and L. Chierchia, "KAM Stability and Celestial Mechanics,", Memoirs American Mathematical Society, 187 (2007).   Google Scholar

[11]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Arch. Rational Mech. Anal., 191 (2009), 311.  doi: 10.1007/s00205-008-0141-5.  Google Scholar

[12]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman),, Ergod. Th. Dynam. Sys., 24 (2004), 1521.   Google Scholar

[13]

A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies,, Celest. Mech. Dyn. Astr., 104 (2009), 159.  doi: 10.1007/s10569-009-9192-7.  Google Scholar

[14]

J. M. Greene, A method for determining a stochastic transition,, J. Math. Phys., 20 (1979), 1183.  doi: 10.1063/1.524170.  Google Scholar

[15]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49.   Google Scholar

[16]

A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964).   Google Scholar

[17]

A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian,, Dokl. Akad. Nauk SSSR, 98 (1954), 527.   Google Scholar

[18]

W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem,, Astron. J., 71 (1966), 568.  doi: 10.1086/109964.  Google Scholar

[19]

À. Jorba and J. Villanueva, Effective stability around periodic orbits of the spatial RTBP,, In: Hamiltonian Systems with Three or More Degrees of Freedom, 533 (1999), 628.   Google Scholar

[20]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian,, Celest. Mech. Dyn. Astr., 62 (1995), 193.  doi: 10.1007/BF00692088.  Google Scholar

[21]

P. Le Calvez, Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff,, Comm. Math. Phys., 106 (1986), 383.  doi: 10.1007/BF01207253.  Google Scholar

[22]

B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem,, Celestial Mechanics, 3 (1971), 408.  doi: 10.1007/BF01227790.  Google Scholar

[23]

R. de la Llave and C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems,, MPEJ, 10 (2004).   Google Scholar

[24]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems,, Celest. Mech. Dyn. Astr., 78 (2000), 47.  doi: 10.1023/A:1011139523256.  Google Scholar

[25]

U. Locatelli and A. Giorgilli, Construction of Kolmogorov's normal form for a planetary system,, Reg. Chaotic Dyn., 10 (2005), 153.  doi: 10.1070/RD2005v010n02ABEH000309.  Google Scholar

[26]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nach. Akad. Wiss. Göttingen, (1962), 1.   Google Scholar

[27]

G. Pinzari, "On the Kolmogorov set for Many-Body Problems,", Ph.D. Thesis, (2009).   Google Scholar

[28]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasi-periodic motions,, Celest. Mech. Dyn. Astr., 62 (1995), 219.  doi: 10.1007/BF00692089.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Uspehi Mat. Nauk, 18 (1963), 91.   Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics,", Encyclopaedia of Mathematical Sciences {\bf 3}, 3 (2006).   Google Scholar

[3]

A. Berretti, A. Celletti, L. Chierchia and C. Falcolini, Natural boundaries for area-preserving twist maps,, J. Stat. Phys., 66 (1992), 1613.  doi: 10.1007/BF01054437.  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]

L. Biasco, L. Chierchia and E. Valdinoci, $N$-dimensional elliptic invariant tori for the planar $(N+1)$-body problem,, SIAM Journal on Mathematical Analysis, 37 (2006), 1560.  doi: 10.1137/S0036141004443646.  Google Scholar

[6]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order amidst Chaos,", Lecture Notes in Mathematics {\bf 1645}, 1645 (1645).   Google Scholar

[7]

H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.  doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[8]

A. Celletti, Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance (Part I),, J. of Applied Math. and Physics (ZAMP), 41 (1990), 174.   Google Scholar

[9]

A. Celletti, Construction of librational invariant tori in the spin-orbit problem,, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61.  doi: 10.1007/BF00942847.  Google Scholar

[10]

A. Celletti and L. Chierchia, "KAM Stability and Celestial Mechanics,", Memoirs American Mathematical Society, 187 (2007).   Google Scholar

[11]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Arch. Rational Mech. Anal., 191 (2009), 311.  doi: 10.1007/s00205-008-0141-5.  Google Scholar

[12]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman),, Ergod. Th. Dynam. Sys., 24 (2004), 1521.   Google Scholar

[13]

A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies,, Celest. Mech. Dyn. Astr., 104 (2009), 159.  doi: 10.1007/s10569-009-9192-7.  Google Scholar

[14]

J. M. Greene, A method for determining a stochastic transition,, J. Math. Phys., 20 (1979), 1183.  doi: 10.1063/1.524170.  Google Scholar

[15]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49.   Google Scholar

[16]

A. Ya. Khinchin, "Continued Fractions,", The University of Chicago Press, (1964).   Google Scholar

[17]

A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian,, Dokl. Akad. Nauk SSSR, 98 (1954), 527.   Google Scholar

[18]

W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem,, Astron. J., 71 (1966), 568.  doi: 10.1086/109964.  Google Scholar

[19]

À. Jorba and J. Villanueva, Effective stability around periodic orbits of the spatial RTBP,, In: Hamiltonian Systems with Three or More Degrees of Freedom, 533 (1999), 628.   Google Scholar

[20]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian,, Celest. Mech. Dyn. Astr., 62 (1995), 193.  doi: 10.1007/BF00692088.  Google Scholar

[21]

P. Le Calvez, Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff,, Comm. Math. Phys., 106 (1986), 383.  doi: 10.1007/BF01207253.  Google Scholar

[22]

B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem,, Celestial Mechanics, 3 (1971), 408.  doi: 10.1007/BF01227790.  Google Scholar

[23]

R. de la Llave and C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems,, MPEJ, 10 (2004).   Google Scholar

[24]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems,, Celest. Mech. Dyn. Astr., 78 (2000), 47.  doi: 10.1023/A:1011139523256.  Google Scholar

[25]

U. Locatelli and A. Giorgilli, Construction of Kolmogorov's normal form for a planetary system,, Reg. Chaotic Dyn., 10 (2005), 153.  doi: 10.1070/RD2005v010n02ABEH000309.  Google Scholar

[26]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nach. Akad. Wiss. Göttingen, (1962), 1.   Google Scholar

[27]

G. Pinzari, "On the Kolmogorov set for Many-Body Problems,", Ph.D. Thesis, (2009).   Google Scholar

[28]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasi-periodic motions,, Celest. Mech. Dyn. Astr., 62 (1995), 219.  doi: 10.1007/BF00692089.  Google Scholar

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