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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

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 & Continuous Dynamical Systems - A, 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). Google Scholar

[2]

V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions,, preprint, (2011). Google Scholar

[3]

V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem,, preprint, (2011). Google Scholar

[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 (). Google Scholar

[5]

S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$,, Mosc. Univ. Mech. Bull., 39 (1984), 24. Google Scholar

[6]

S. V. Bolotin and P. Negrini, Regularization and topological entropy for the spatial $n$-center problem,, Ergodic Theory Dyn. Systems, 21 (2001), 383. doi: 10.1017/S0143385701001195. Google Scholar

[7]

S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem,, J. Differential Equations, 190 (2003), 539. Google Scholar

[8]

H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,", Colletion Mathématiques Appliquées por la Maîtrise, (1983). Google Scholar

[9]

R. Castelli, "On the Variational Approach to the One and N-Centre Problem with Weak Forces,", Ph.D Thesis, (2009). Google Scholar

[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. doi: 10.4007/annals.2008.167.325. Google Scholar

[11]

K.-C. Chen, Variational constructions for some satellite orbits in periodic gravitational force fields,, Amer. J. Math., 132 (2010), 681. doi: 10.1353/ajm.0.0124. Google Scholar

[12]

L. Dimare, Chaotic quasi-collision trajectories in the $3$-centre problem,, Celest. Mech Dyn. Astr., 107 (2010), 427. doi: 10.1007/s10569-010-9284-4. Google Scholar

[13]

M. P. Do Carmo, "Riemaniann Geometry,", Series of Mathematics, (1992). Google Scholar

[14]

P. Felmer and K. Tanaka, Scattering solutions for planar singular Hamiltonian systems via minimization,, Adv. Differential Equations, 5 (2000), 1519. Google Scholar

[15]

D. L. Ferrario, Transitive decomposition of symmetry groups for the n-body problem,, Adv. Math., 213 (2007), 763. doi: 10.1016/j.aim.2007.01.009. Google Scholar

[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. doi: 10.1007/s00222-010-0306-3. Google Scholar

[17]

M. Klein and A. Knauf, "Classical Planar Scattering by Coulombic Potentials,", Lecture Notes in Physics, (1992). Google Scholar

[18]

A. Knauf, The n-centre problem of celestial mechanics for large energies,, J. Eur. Math. Soc., 4 (2002), 1. doi: 10.1007/s100970100037. Google Scholar

[19]

A. Knauf and M. Krapf, The escape rate of a molecule,, Math. Phys. Anal. Geom., 13 (2010), 159. doi: 10.1007/s11040-010-9073-z. Google Scholar

[20]

A. Knauf and I. A. Taimanov, On the integrability of the $n$-centre problem,, Math. Ann., 331 (2005), 631. doi: 10.1007/s00208-004-0598-y. Google Scholar

[21]

T. Levi-Civita, Sur la régularisation du problème des trois corps,, Acta Math., 42 (1920), 99. doi: 10.1007/BF02404404. Google Scholar

[22]

C. Marchal, How the method of minimization of action avoids singularities,, Cel. Mech. Dyn. Ast., 83 (2002), 325. doi: 10.1023/A:1020128408706. Google Scholar

[23]

C. Moore, Braids in classical dynamics,, Phys. Rev. Lett., 70 (1993), 3675. doi: 10.1103/PhysRevLett.70.3675. Google Scholar

[24]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure. Appl. Math., 23 (1970), 609. doi: 10.1002/cpa.3160230406. Google Scholar

[25]

H. Seifert, Periodischer bewegungen mechanischer system,, Math. Zeit, 51 (1948), 197. doi: 10.1007/BF01291002. Google Scholar

[26]

S. Terracini and A. Venturelli, Symmetric trajectories for the $2N$-body problem with equal masses,, Arch. Ration. Mech. Anal., 184 (2007), 465. doi: 10.1007/s00205-006-0030-8. Google Scholar

[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. Google Scholar

[28]

A. Venturelli, "Application de la Minimisation de l'Action au Problème de N Corps Dans le Plan e Dans lÉspace,", Ph.D Thesis, (2002). Google Scholar

[29]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton University Press, (1941). Google Scholar

[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, (1959). Google Scholar

show all references

References:
[1]

A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems,", Birkhäuser, (1993). Google Scholar

[2]

V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions,, preprint, (2011). Google Scholar

[3]

V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem,, preprint, (2011). Google Scholar

[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 (). Google Scholar

[5]

S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$,, Mosc. Univ. Mech. Bull., 39 (1984), 24. Google Scholar

[6]

S. V. Bolotin and P. Negrini, Regularization and topological entropy for the spatial $n$-center problem,, Ergodic Theory Dyn. Systems, 21 (2001), 383. doi: 10.1017/S0143385701001195. Google Scholar

[7]

S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem,, J. Differential Equations, 190 (2003), 539. Google Scholar

[8]

H. Brezis, "Analyse Fonctionnelle, Théorie et Applications,", Colletion Mathématiques Appliquées por la Maîtrise, (1983). Google Scholar

[9]

R. Castelli, "On the Variational Approach to the One and N-Centre Problem with Weak Forces,", Ph.D Thesis, (2009). Google Scholar

[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. doi: 10.4007/annals.2008.167.325. Google Scholar

[11]

K.-C. Chen, Variational constructions for some satellite orbits in periodic gravitational force fields,, Amer. J. Math., 132 (2010), 681. doi: 10.1353/ajm.0.0124. Google Scholar

[12]

L. Dimare, Chaotic quasi-collision trajectories in the $3$-centre problem,, Celest. Mech Dyn. Astr., 107 (2010), 427. doi: 10.1007/s10569-010-9284-4. Google Scholar

[13]

M. P. Do Carmo, "Riemaniann Geometry,", Series of Mathematics, (1992). Google Scholar

[14]

P. Felmer and K. Tanaka, Scattering solutions for planar singular Hamiltonian systems via minimization,, Adv. Differential Equations, 5 (2000), 1519. Google Scholar

[15]

D. L. Ferrario, Transitive decomposition of symmetry groups for the n-body problem,, Adv. Math., 213 (2007), 763. doi: 10.1016/j.aim.2007.01.009. Google Scholar

[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. doi: 10.1007/s00222-010-0306-3. Google Scholar

[17]

M. Klein and A. Knauf, "Classical Planar Scattering by Coulombic Potentials,", Lecture Notes in Physics, (1992). Google Scholar

[18]

A. Knauf, The n-centre problem of celestial mechanics for large energies,, J. Eur. Math. Soc., 4 (2002), 1. doi: 10.1007/s100970100037. Google Scholar

[19]

A. Knauf and M. Krapf, The escape rate of a molecule,, Math. Phys. Anal. Geom., 13 (2010), 159. doi: 10.1007/s11040-010-9073-z. Google Scholar

[20]

A. Knauf and I. A. Taimanov, On the integrability of the $n$-centre problem,, Math. Ann., 331 (2005), 631. doi: 10.1007/s00208-004-0598-y. Google Scholar

[21]

T. Levi-Civita, Sur la régularisation du problème des trois corps,, Acta Math., 42 (1920), 99. doi: 10.1007/BF02404404. Google Scholar

[22]

C. Marchal, How the method of minimization of action avoids singularities,, Cel. Mech. Dyn. Ast., 83 (2002), 325. doi: 10.1023/A:1020128408706. Google Scholar

[23]

C. Moore, Braids in classical dynamics,, Phys. Rev. Lett., 70 (1993), 3675. doi: 10.1103/PhysRevLett.70.3675. Google Scholar

[24]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure. Appl. Math., 23 (1970), 609. doi: 10.1002/cpa.3160230406. Google Scholar

[25]

H. Seifert, Periodischer bewegungen mechanischer system,, Math. Zeit, 51 (1948), 197. doi: 10.1007/BF01291002. Google Scholar

[26]

S. Terracini and A. Venturelli, Symmetric trajectories for the $2N$-body problem with equal masses,, Arch. Ration. Mech. Anal., 184 (2007), 465. doi: 10.1007/s00205-006-0030-8. Google Scholar

[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. Google Scholar

[28]

A. Venturelli, "Application de la Minimisation de l'Action au Problème de N Corps Dans le Plan e Dans lÉspace,", Ph.D Thesis, (2002). Google Scholar

[29]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton University Press, (1941). Google Scholar

[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, (1959). Google Scholar

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