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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

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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99.  Google Scholar

[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.  Google Scholar

[8]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.  Google Scholar

[12]

R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi: 10.1007/BF02566226.  Google Scholar

[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.  Google Scholar

[14]

C. L. Siegel and J. K. Moser, "Lectures on Celestial Mechanics," Classics in Mathematics, Springer-Verlag, Berlin, 1995. Google Scholar

[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.  Google Scholar

[16]

V. G. Szebehely, "Theory of Orbits -- The Restricted Problem of Three Bodies," Academic Press, New York, 1967. Google Scholar

[17]

J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932. Google Scholar

[18]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," 4th edition, Cambridge University Press, New York, 1959. Google Scholar

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99.  Google Scholar

[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.  Google Scholar

[8]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math., 42 (1920), 99-144. doi: 10.1007/BF02404404.  Google Scholar

[12]

R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi: 10.1007/BF02566226.  Google Scholar

[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.  Google Scholar

[14]

C. L. Siegel and J. K. Moser, "Lectures on Celestial Mechanics," Classics in Mathematics, Springer-Verlag, Berlin, 1995. Google Scholar

[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.  Google Scholar

[16]

V. G. Szebehely, "Theory of Orbits -- The Restricted Problem of Three Bodies," Academic Press, New York, 1967. Google Scholar

[17]

J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932. Google Scholar

[18]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies," 4th edition, Cambridge University Press, New York, 1959. Google Scholar

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