-
Previous Article
Traveling waves of diffusive predator-prey systems: Disease outbreak propagation
- DCDS Home
- This Issue
-
Next Article
Adaptation of an ecological territorial model to street gang spatial patterns in Los Angeles
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 |
[1] |
Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 |
[2] |
Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 |
[3] |
Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1 |
[4] |
Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 |
[5] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[6] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[7] |
Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 |
[8] |
Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 |
[9] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[10] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[11] |
Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020401 |
[12] |
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 |
[13] |
Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 |
[14] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[15] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378 |
[16] |
Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 |
[17] |
Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 |
[18] |
Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 |
[19] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]