# American Institute of Mathematical Sciences

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), 91-192.  Google Scholar [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Encyclopaedia of Mathematical Sciences 3, Dynamical Systems III, Third edition, Springer-Verlag, Berlin, 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), 1613-1630. 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), 91-135. 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), 1560-1588. 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 1645, Springer-Verlag, Berlin, 1996.  Google Scholar [7] H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. 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), 174-204.  Google Scholar [9] A. Celletti, Construction of librational invariant tori in the spin-orbit problem, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61-80. 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), 311-345. 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), 1521-1582.  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), 159-173. 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), 1183-1201. doi: 10.1063/1.524170.  Google Scholar [15] M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, fasc. 2, 3 (1966), 49-66. Google Scholar [16] A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 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), 527-530.  Google Scholar [18] W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem, Astron. J., 71 (1966), 568-578. 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, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 533, C. Simó ed., Kluwer Acad. Publ., Dordrecht, (1999), 628-632.  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), 193-217. 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), 383-394. doi: 10.1007/BF01207253.  Google Scholar [22] B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem, Celestial Mechanics, 3 (1971), 408-426. 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), 47-74. 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), 153-171. doi: 10.1070/RD2005v010n02ABEH000309.  Google Scholar [26] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, (1962), 1-20.  Google Scholar [27] G. Pinzari, "On the Kolmogorov set for Many-Body Problems," Ph.D. Thesis, Università Roma Tre 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), 219-261. doi: 10.1007/BF00692089.  Google Scholar

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##### References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192.  Google Scholar [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Encyclopaedia of Mathematical Sciences 3, Dynamical Systems III, Third edition, Springer-Verlag, Berlin, 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), 1613-1630. 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), 91-135. 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), 1560-1588. 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 1645, Springer-Verlag, Berlin, 1996.  Google Scholar [7] H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. 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), 174-204.  Google Scholar [9] A. Celletti, Construction of librational invariant tori in the spin-orbit problem, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61-80. 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), 311-345. 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), 1521-1582.  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), 159-173. 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), 1183-1201. doi: 10.1063/1.524170.  Google Scholar [15] M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, fasc. 2, 3 (1966), 49-66. Google Scholar [16] A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 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), 527-530.  Google Scholar [18] W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem, Astron. J., 71 (1966), 568-578. 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, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 533, C. Simó ed., Kluwer Acad. Publ., Dordrecht, (1999), 628-632.  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), 193-217. 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), 383-394. doi: 10.1007/BF01207253.  Google Scholar [22] B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem, Celestial Mechanics, 3 (1971), 408-426. 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), 47-74. 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), 153-171. doi: 10.1070/RD2005v010n02ABEH000309.  Google Scholar [26] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, (1962), 1-20.  Google Scholar [27] G. Pinzari, "On the Kolmogorov set for Many-Body Problems," Ph.D. Thesis, Università Roma Tre 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), 219-261. doi: 10.1007/BF00692089.  Google Scholar
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