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Symbolic dynamics for the $N$-centre problem at negative energies
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 |
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 ().
|
[5] |
S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$,, Mosc. Univ. Mech. Bull., 39 (1984), 24.
|
[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. |
[7] |
S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem,, J. Differential Equations, 190 (2003), 539.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
T. Levi-Civita, Sur la régularisation du problème des trois corps,, Acta Math., 42 (1920), 99.
doi: 10.1007/BF02404404. |
[22] |
C. Marchal, How the method of minimization of action avoids singularities,, Cel. Mech. Dyn. Ast., 83 (2002), 325.
doi: 10.1023/A:1020128408706. |
[23] |
C. Moore, Braids in classical dynamics,, Phys. Rev. Lett., 70 (1993), 3675.
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.
doi: 10.1002/cpa.3160230406. |
[25] |
H. Seifert, Periodischer bewegungen mechanischer system,, Math. Zeit, 51 (1948), 197.
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.
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.
|
[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 ().
|
[5] |
S. V. Bolotin, Nonintegrability of the $n$-center problem for $n>2$,, Mosc. Univ. Mech. Bull., 39 (1984), 24.
|
[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. |
[7] |
S. V. Bolotin and P. Negrini, Chaotic behaviour in the $3$-center problem,, J. Differential Equations, 190 (2003), 539.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
T. Levi-Civita, Sur la régularisation du problème des trois corps,, Acta Math., 42 (1920), 99.
doi: 10.1007/BF02404404. |
[22] |
C. Marchal, How the method of minimization of action avoids singularities,, Cel. Mech. Dyn. Ast., 83 (2002), 325.
doi: 10.1023/A:1020128408706. |
[23] |
C. Moore, Braids in classical dynamics,, Phys. Rev. Lett., 70 (1993), 3675.
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.
doi: 10.1002/cpa.3160230406. |
[25] |
H. Seifert, Periodischer bewegungen mechanischer system,, Math. Zeit, 51 (1948), 197.
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.
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.
|
[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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