On the regularization of the collision solutions of the one-center problem with weak forces

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December 2011, 31(4): 1197-1218. doi: 10.3934/dcds.2011.31.1197

On the regularization of the collision solutions of the one-center problem with weak forces

Roberto Castelli ^1, and Susanna Terracini ^2,

1.	BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park, 48160 Derio, Bizkaia,, Spain
2.	Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano

Received January 2010 Revised March 2010 Published September 2011

Abstract
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We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with transmission trajectories, is continuous, though not differentiable, with respect to the initial data.

Keywords: regularization technique, Two-body problem, weak singular potential., logarithmic potential.

Mathematics Subject Classification: Primary: 70F16, 70F35; Secondary: 34H4.

Citation: Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1197-1218. doi: 10.3934/dcds.2011.31.1197

References:

[1]	S. J. Aarseth, Dynamical evolution of clusters of galaxies I, Monthly Notices of the Royal Astronomical Society, 126 (1963), 223-255.
[2]	V. Barutello, D. L. 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, 78 pp.
[3]	G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two-body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi: 10.3934/cpaa.2003.2.323.
[4]	E. De Giorgi, Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. doi: 10.1215/S0012-7094-96-08114-4.
[5]	C. C. Dyer and P. S. S. Ip, Softening in N-body simulations of collisionless systems, Astrophysical Journal, 409 (1993), 60-67. doi: 10.1086/172641.
[6]	R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99.
[7]	D. L. 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.
[8]	W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.
[9]	L. Hernquist and J. E. Barnes, Are some n-body algorithms intrinsically less collisional than others?, Astrophysical Journal, 349 (1990), 562-569. doi: 10.1086/168343.
[10]	P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi: 10.1515/crll.1965.218.204.
[11]	T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.
[12]	R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi: 10.1007/BF02566226.
[13]	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.
[14]	C. L. Siegel and J. K. Moser, "Lectures on Celestial Mechanics," Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[15]	C. Stoica and A. Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi: 10.1088/0305-4470/36/28/302.
[16]	V. G. Szebehely, "Theory of Orbits -- The Restricted Problem of Three Bodies," Academic Press, New York, 1967.
[17]	J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932.
[18]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," 4^th edition, Cambridge University Press, New York, 1959.

show all references

References:

[1]	S. J. Aarseth, Dynamical evolution of clusters of galaxies I, Monthly Notices of the Royal Astronomical Society, 126 (1963), 223-255.
[2]	V. Barutello, D. L. 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, 78 pp.
[3]	G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two-body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi: 10.3934/cpaa.2003.2.323.
[4]	E. De Giorgi, Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. doi: 10.1215/S0012-7094-96-08114-4.
[5]	C. C. Dyer and P. S. S. Ip, Softening in N-body simulations of collisionless systems, Astrophysical Journal, 409 (1993), 60-67. doi: 10.1086/172641.
[6]	R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99.
[7]	D. L. 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.
[8]	W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.
[9]	L. Hernquist and J. E. Barnes, Are some n-body algorithms intrinsically less collisional than others?, Astrophysical Journal, 349 (1990), 562-569. doi: 10.1086/168343.
[10]	P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi: 10.1515/crll.1965.218.204.
[11]	T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.
[12]	R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi: 10.1007/BF02566226.
[13]	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.
[14]	C. L. Siegel and J. K. Moser, "Lectures on Celestial Mechanics," Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[15]	C. Stoica and A. Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi: 10.1088/0305-4470/36/28/302.
[16]	V. G. Szebehely, "Theory of Orbits -- The Restricted Problem of Three Bodies," Academic Press, New York, 1967.
[17]	J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932.
[18]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," 4^th edition, Cambridge University Press, New York, 1959.

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