Symbolic dynamics for the N-centre problem at negative energies

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September 2012, 32(9): 3245-3301. doi: 10.3934/dcds.2012.32.3245

Symbolic dynamics for the $N$-centre problem at negative energies

Nicola Soave ^1, and Susanna Terracini ^2,

1.	Università di Milano Bicocca - Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy
2.	Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano

Received February 2012 Revised March 2012 Published April 2012

Abstract
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We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\alpha < 0$, $\alpha \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.

Keywords: chaotic motions, $N$-centre problem, symbolic dynamics, Levi-Civita regularization..

Mathematics Subject Classification: Primary: 70F10, 37N05; Secondary: 70F15, 37J3.

Citation: Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245

References:

[1]	A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems," Birkhäuser, 1993.
[2]	V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, preprint, 2011.
[3]	V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, preprint, 2011.
[4]	V. Barutello, D. L. Ferrario and S. Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Notices IMRN, 2008, Art. ID rnn 069, 78 pp.
[5]	S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$, Mosc. Univ. Mech. Bull., 39 (1984), 24-28; translated from Vestnik Mosk. Univ. Ser. I Math. Mekh., 1984, 65-68.
[6]	S. V. Bolotin and P. Negrini, Regularization and topological entropy for the spatial $n$-center problem, Ergodic Theory Dyn. Systems, 21 (2001), 383-399. doi: 10.1017/S0143385701001195.
[7]	S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem, J. Differential Equations, 190 (2003), 539-558.
[8]	H. Brezis, "Analyse Fonctionnelle, Théorie et Applications," Colletion Mathématiques Appliquées por la Maîtrise, Massons, Paris, 1983.
[9]	R. Castelli, "On the Variational Approach to the One and N-Centre Problem with Weak Forces," Ph.D Thesis, University of Milano-Bicocca, 2009.
[10]	K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Ann. of Math. (2), 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.
[11]	K.-C. Chen, Variational constructions for some satellite orbits in periodic gravitational force fields, Amer. J. Math., 132 (2010), 681-709. doi: 10.1353/ajm.0.0124.
[12]	L. Dimare, Chaotic quasi-collision trajectories in the $3$-centre problem, Celest. Mech Dyn. Astr., 107 (2010), 427-449. doi: 10.1007/s10569-010-9284-4.
[13]	M. P. Do Carmo, "Riemaniann Geometry," Series of Mathematics, Birkhäuser, Boston, 1992.
[14]	P. Felmer and K. Tanaka, Scattering solutions for planar singular Hamiltonian systems via minimization, Adv. Differential Equations, 5 (2000), 1519-1544.
[15]	D. L. Ferrario, Transitive decomposition of symmetry groups for the n-body problem, Adv. Math., 213 (2007), 763-784. doi: 10.1016/j.aim.2007.01.009.
[16]	G. Fusco, G. F. Gronchi and P. Negrini, Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem, Inv. Math.,185 (2011), 283-332. doi: 10.1007/s00222-010-0306-3.
[17]	M. Klein and A. Knauf, "Classical Planar Scattering by Coulombic Potentials," Lecture Notes in Physics, Springer, Berlin, 1992.
[18]	A. Knauf, The n-centre problem of celestial mechanics for large energies, J. Eur. Math. Soc., 4 (2002), 1-114. doi: 10.1007/s100970100037.
[19]	A. Knauf and M. Krapf, The escape rate of a molecule, Math. Phys. Anal. Geom., 13 (2010), 159-189. doi: 10.1007/s11040-010-9073-z.
[20]	A. Knauf and I. A. Taimanov, On the integrability of the $n$-centre problem, Math. Ann., 331 (2005), 631-649. doi: 10.1007/s00208-004-0598-y.
[21]	T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.
[22]	C. Marchal, How the method of minimization of action avoids singularities, Cel. Mech. Dyn. Ast., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.
[23]	C. Moore, Braids in classical dynamics, Phys. Rev. Lett., 70 (1993), 3675-3679. doi: 10.1103/PhysRevLett.70.3675.
[24]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure. Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[25]	H. Seifert, Periodischer bewegungen mechanischer system, Math. Zeit, 51 (1948), 197-216. doi: 10.1007/BF01291002.
[26]	S. Terracini and A. Venturelli, Symmetric trajectories for the $2N$-body problem with equal masses, Arch. Ration. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.
[27]	A. Venturelli, Une caractérisation variationelle des solutions de Lagrange du probl\`eme plan des trois corps, Comp. Rend. Acad. Sci. Paris Sér. I Math., 332 (2001), 641-644.
[28]	A. Venturelli, "Application de la Minimisation de l'Action au Problème de N Corps Dans le Plan e Dans lÉspace," Ph.D Thesis, University Paris VII, 2002.
[29]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton University Press, 1941.
[30]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," Cambridge University Press, New York, 1959.

show all references

References:

[1]	A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems," Birkhäuser, 1993.
[2]	V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, preprint, 2011.
[3]	V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, preprint, 2011.
[4]	V. Barutello, D. L. Ferrario and S. Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Notices IMRN, 2008, Art. ID rnn 069, 78 pp.
[5]	S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$, Mosc. Univ. Mech. Bull., 39 (1984), 24-28; translated from Vestnik Mosk. Univ. Ser. I Math. Mekh., 1984, 65-68.
[6]	S. V. Bolotin and P. Negrini, Regularization and topological entropy for the spatial $n$-center problem, Ergodic Theory Dyn. Systems, 21 (2001), 383-399. doi: 10.1017/S0143385701001195.
[7]	S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem, J. Differential Equations, 190 (2003), 539-558.
[8]	H. Brezis, "Analyse Fonctionnelle, Théorie et Applications," Colletion Mathématiques Appliquées por la Maîtrise, Massons, Paris, 1983.
[9]	R. Castelli, "On the Variational Approach to the One and N-Centre Problem with Weak Forces," Ph.D Thesis, University of Milano-Bicocca, 2009.
[10]	K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Ann. of Math. (2), 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.
[11]	K.-C. Chen, Variational constructions for some satellite orbits in periodic gravitational force fields, Amer. J. Math., 132 (2010), 681-709. doi: 10.1353/ajm.0.0124.
[12]	L. Dimare, Chaotic quasi-collision trajectories in the $3$-centre problem, Celest. Mech Dyn. Astr., 107 (2010), 427-449. doi: 10.1007/s10569-010-9284-4.
[13]	M. P. Do Carmo, "Riemaniann Geometry," Series of Mathematics, Birkhäuser, Boston, 1992.
[14]	P. Felmer and K. Tanaka, Scattering solutions for planar singular Hamiltonian systems via minimization, Adv. Differential Equations, 5 (2000), 1519-1544.
[15]	D. L. Ferrario, Transitive decomposition of symmetry groups for the n-body problem, Adv. Math., 213 (2007), 763-784. doi: 10.1016/j.aim.2007.01.009.
[16]	G. Fusco, G. F. Gronchi and P. Negrini, Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem, Inv. Math.,185 (2011), 283-332. doi: 10.1007/s00222-010-0306-3.
[17]	M. Klein and A. Knauf, "Classical Planar Scattering by Coulombic Potentials," Lecture Notes in Physics, Springer, Berlin, 1992.
[18]	A. Knauf, The n-centre problem of celestial mechanics for large energies, J. Eur. Math. Soc., 4 (2002), 1-114. doi: 10.1007/s100970100037.
[19]	A. Knauf and M. Krapf, The escape rate of a molecule, Math. Phys. Anal. Geom., 13 (2010), 159-189. doi: 10.1007/s11040-010-9073-z.
[20]	A. Knauf and I. A. Taimanov, On the integrability of the $n$-centre problem, Math. Ann., 331 (2005), 631-649. doi: 10.1007/s00208-004-0598-y.
[21]	T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.
[22]	C. Marchal, How the method of minimization of action avoids singularities, Cel. Mech. Dyn. Ast., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.
[23]	C. Moore, Braids in classical dynamics, Phys. Rev. Lett., 70 (1993), 3675-3679. doi: 10.1103/PhysRevLett.70.3675.
[24]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure. Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[25]	H. Seifert, Periodischer bewegungen mechanischer system, Math. Zeit, 51 (1948), 197-216. doi: 10.1007/BF01291002.
[26]	S. Terracini and A. Venturelli, Symmetric trajectories for the $2N$-body problem with equal masses, Arch. Ration. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.
[27]	A. Venturelli, Une caractérisation variationelle des solutions de Lagrange du probl\`eme plan des trois corps, Comp. Rend. Acad. Sci. Paris Sér. I Math., 332 (2001), 641-644.
[28]	A. Venturelli, "Application de la Minimisation de l'Action au Problème de N Corps Dans le Plan e Dans lÉspace," Ph.D Thesis, University Paris VII, 2002.
[29]	A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton University Press, 1941.
[30]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," Cambridge University Press, New York, 1959.

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