# American Institute of Mathematical Sciences

January  2021, 41(1): 61-86. doi: 10.3934/dcds.2020218

## Minimal collision arcs asymptotic to central configurations

 Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Received  January 2020 Revised  March 2020 Published  May 2020

We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree $-\alpha$, with $\alpha\in(0,2)$ and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the collision manifold having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

Citation: Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
##### References:
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Terracini, An Index theory for asymptotic motions under singular potentials, preprint 2018, arXiv: 1705.01291. Google Scholar [6] V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, Arch. Ration. Mech. Anal., 207 (2013), 583-609.  doi: 10.1007/s00205-012-0565-9.  Google Scholar [7] V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, Calc. Var. Partial Differential Equations, 49 (2014), 391-429.  doi: 10.1007/s00526-012-0587-z.  Google Scholar [8] A. Boscaggin, W. Dambrosio and D. Papini, Parabolic solutions for the planar $N$-centre problem: Multiplicity and scattering, Ann. Mat. Pura Appl. (4), 197 (2018), 869-882.  doi: 10.1007/s10231-017-0707-7.  Google Scholar [9] A. Boscaggin, W. Dambrosio and S. Terracini, Scattering parabolic solutions for the spatial $N$-centre problem, Arch. Ration. Mech. Anal., 223 (2017), 1269-1306.  doi: 10.1007/s00205-016-1057-0.  Google Scholar [10] R. Devaney, Collision orbits in the anisotropic Kepler problem, Invent. Math., 45 (1978), 221-251.  doi: 10.1007/BF01403170.  Google Scholar [11] R. Devaney, Singularities in classical mechanical systems, in Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979–80), vol. 10 of Progr. Math., Birkhäuser Boston, Mass., 1981,211–333. doi: 10.1007/978-1-4899-6696-4_7.  Google Scholar [12] R. Devaney, Blowing up singularities in classical mechanical systems, Amer. Math. Monthly, 89 (1982), 535-552.  doi: 10.1080/00029890.1982.11995493.  Google Scholar [13] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [14] M. Gutzwiller, The anisotropic Kepler problem in two dimensions, J. Mathematical Phys., 14 (1973), 139-152.  doi: 10.1063/1.1666164.  Google Scholar [15] M. Gutzwiller, Bernoulli sequences and trajectories in the anisotropic Kepler problem, J. Mathematical Phys., 18 (1977), 806-823.  doi: 10.1063/1.523310.  Google Scholar [16] M. Gutzwiller, Periodic orbits in the anisotropic Kepler problem, in Classical Mechanics and Dynamical Systems (Medford, Mass., 1979), vol. 70 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1981, 69–90.  Google Scholar [17] M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013. doi: 10.1016/C2009-0-61160-0.  Google Scholar [18] X. Hu and G. Yu, An index theory for zero energy solutions of the planar anisotropic kepler problem, Communications in Mathematical Physics, 361 (2018), 709-736.  doi: 10.1007/s00220-018-3184-y.  Google Scholar [19] N. Hulkower and D. Saari, On the manifolds of total collapse orbits and of completely parabolic orbits for the $n$-body problem, J. Differential Equations, 41 (1981), 27-43.  doi: 10.1016/0022-0396(81)90051-6.  Google Scholar [20] A. Knauf, Mathematical Physics: Classical Mechanics, vol. 109 of Unitext, Springer-Verlag, Berlin, 2018, Translated from the 2017 second German edition by Jochen Denzler. doi: 10.1007/978-3-662-55774-7.  Google Scholar [21] E. Maderna and A. Venturelli, Globally minimizing parabolic motions in the Newtonian $N$-body problem, Arch. Ration. Mech. Anal., 194 (2009), 283-313.  doi: 10.1007/s00205-008-0175-8.  Google Scholar [22] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  doi: 10.1007/BF01390175.  Google Scholar [23] R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557.   Google Scholar [24] R. Moeckel, R. Montgomery and A. Venturelli, From brake to syzygy, Arch. Ration. Mech. Anal., 204 (2012), 1009-1060.  doi: 10.1007/s00205-012-0502-y.  Google Scholar [25] K. Sundman, Mémoire sur le problème des trois corps, Acta Math., 36 (1913), 105-179.  doi: 10.1007/BF02422379.  Google Scholar [26] G. Teschl, Ordinary Differential Equations and Dynamical Systems, vol. 140 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.  Google Scholar [27] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, v. 5, Princeton University Press, Princeton, N. J., 1941.  Google Scholar

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##### References:
 [1] A. Ambrosetti and V. Coti Zelati, Periodic solutions of singular Lagrangian systems, Progress in Nonlinear Differential Equations and their Applications, 10, Birkhäuser Boston Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar [2] V. Arnol'd, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, [1989], Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [3] V. Barutello, G. Canneori and S. Terracini, Symbolic dynamics for the anisotropic $N$-centre problem, in preparation, 2020. Google Scholar [4] V. Barutello, D. Ferrario and S. Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Not. IMRN, 2008, Art. ID rnn 069, 78pp.  Google Scholar [5] V. Barutello, X. Hu, A. Portaluri and S. Terracini, An Index theory for asymptotic motions under singular potentials, preprint 2018, arXiv: 1705.01291. Google Scholar [6] V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, Arch. Ration. Mech. Anal., 207 (2013), 583-609.  doi: 10.1007/s00205-012-0565-9.  Google Scholar [7] V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, Calc. Var. Partial Differential Equations, 49 (2014), 391-429.  doi: 10.1007/s00526-012-0587-z.  Google Scholar [8] A. Boscaggin, W. Dambrosio and D. Papini, Parabolic solutions for the planar $N$-centre problem: Multiplicity and scattering, Ann. Mat. Pura Appl. (4), 197 (2018), 869-882.  doi: 10.1007/s10231-017-0707-7.  Google Scholar [9] A. Boscaggin, W. Dambrosio and S. Terracini, Scattering parabolic solutions for the spatial $N$-centre problem, Arch. Ration. Mech. Anal., 223 (2017), 1269-1306.  doi: 10.1007/s00205-016-1057-0.  Google Scholar [10] R. Devaney, Collision orbits in the anisotropic Kepler problem, Invent. Math., 45 (1978), 221-251.  doi: 10.1007/BF01403170.  Google Scholar [11] R. Devaney, Singularities in classical mechanical systems, in Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979–80), vol. 10 of Progr. Math., Birkhäuser Boston, Mass., 1981,211–333. doi: 10.1007/978-1-4899-6696-4_7.  Google Scholar [12] R. Devaney, Blowing up singularities in classical mechanical systems, Amer. Math. Monthly, 89 (1982), 535-552.  doi: 10.1080/00029890.1982.11995493.  Google Scholar [13] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [14] M. Gutzwiller, The anisotropic Kepler problem in two dimensions, J. Mathematical Phys., 14 (1973), 139-152.  doi: 10.1063/1.1666164.  Google Scholar [15] M. Gutzwiller, Bernoulli sequences and trajectories in the anisotropic Kepler problem, J. Mathematical Phys., 18 (1977), 806-823.  doi: 10.1063/1.523310.  Google Scholar [16] M. Gutzwiller, Periodic orbits in the anisotropic Kepler problem, in Classical Mechanics and Dynamical Systems (Medford, Mass., 1979), vol. 70 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1981, 69–90.  Google Scholar [17] M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013. doi: 10.1016/C2009-0-61160-0.  Google Scholar [18] X. Hu and G. Yu, An index theory for zero energy solutions of the planar anisotropic kepler problem, Communications in Mathematical Physics, 361 (2018), 709-736.  doi: 10.1007/s00220-018-3184-y.  Google Scholar [19] N. Hulkower and D. Saari, On the manifolds of total collapse orbits and of completely parabolic orbits for the $n$-body problem, J. Differential Equations, 41 (1981), 27-43.  doi: 10.1016/0022-0396(81)90051-6.  Google Scholar [20] A. Knauf, Mathematical Physics: Classical Mechanics, vol. 109 of Unitext, Springer-Verlag, Berlin, 2018, Translated from the 2017 second German edition by Jochen Denzler. doi: 10.1007/978-3-662-55774-7.  Google Scholar [21] E. Maderna and A. Venturelli, Globally minimizing parabolic motions in the Newtonian $N$-body problem, Arch. Ration. Mech. Anal., 194 (2009), 283-313.  doi: 10.1007/s00205-008-0175-8.  Google Scholar [22] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  doi: 10.1007/BF01390175.  Google Scholar [23] R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557.   Google Scholar [24] R. Moeckel, R. Montgomery and A. Venturelli, From brake to syzygy, Arch. Ration. Mech. Anal., 204 (2012), 1009-1060.  doi: 10.1007/s00205-012-0502-y.  Google Scholar [25] K. Sundman, Mémoire sur le problème des trois corps, Acta Math., 36 (1913), 105-179.  doi: 10.1007/BF02422379.  Google Scholar [26] G. Teschl, Ordinary Differential Equations and Dynamical Systems, vol. 140 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.  Google Scholar [27] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, v. 5, Princeton University Press, Princeton, N. J., 1941.  Google Scholar
The local stable manifold characterized in Remark 3.3
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