Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-30T23:55:36.875Z Has data issue: false hasContentIssue false
  • English
  • Français

On weak KAM theory for N-body problems

Published online by Cambridge University Press:  27 April 2011

EZEQUIEL MADERNA
EZEQUIEL MADERNA*
Affiliation:
CMAT, Centro de Matemática, Facultad de Ciencias, Universidad de la República, Montevideo, Uruguay (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 1019 - 1041
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bardi, M. and Cappuzzo-Dolceta, I.. Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations (Systems & Control: Foundations & Applications). Birkhäuser Boston Inc., Boston, MA, 1997.CrossRefGoogle Scholar
[2]Barles, G.. Solutions de viscosité des équations de Hamilton–Jacobi (Mathématiques et Applications, 17). Springer, Paris, 1994.Google Scholar
[3]Chenciner, A.. Action minimizing solutions of the Newtonian N-body problem: from homology to symmetry. Proceedings ICM 2002, Vol. III. Higher Ed. Press, Beijing, 2002, pp. 279–294.Google Scholar
[4]Chenciner, A. and Montgomery, R.. A remarkable periodic solution of the three body problem in the case of equal masses. Ann. of Math. (2) 152 (2000), 881–901.CrossRefGoogle Scholar
[5]Contreras, G.. Action potential and weak KAM solutions. Calc. Var. Partial Differential Equations 13(4) (2001), 427–458.CrossRefGoogle Scholar
[6]Contreras, G., Iturriaga, R., Paternain, G. P. and Paternain, M.. Lagrangian graphs, minimizing measures and Mañé’s critical values. Geom. Funct. Anal. 8(5) (1998), 788–809.CrossRefGoogle Scholar
[7]Da Luz, A. and Maderna, E.. On the dynamics of free time minimizers of the Newtonian N-body problem. Math. Proc. Cambridge Philos. Soc. to appear.Google Scholar
[8]Dugundji, J.. Topology. Allyn and Bacon Inc., Boston, 1970.Google Scholar
[9]Evans, L. C.. Partial Differential Equations (Graduate Studies in Mathematics, 19). American Mathematical Society, Providence, RI, 1998.Google Scholar
[10]Fathi, A.. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris 324(9) (1997), 1043–1046.CrossRefGoogle Scholar
[11]Fathi, A. and Maderna, E.. Weak KAM theorem on non compact manifolds. NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1–27.CrossRefGoogle Scholar
[12]Ferrario, D. L. and Terracini, S.. On the existence of collisionless equivariant minimizers for the classical N-body problem. Invent. Math. 155(2) (2004), 305–362.CrossRefGoogle Scholar
[13]Maderna, E.. Invariance of global solutions of the Hamilton–Jacobi equation. Bull. Soc. Math. France 130(4) (2002), 493–506.CrossRefGoogle Scholar
[14]Maderna, E. and Venturelli, A.. Globally minimizing parabolic motions in the Newtonian N-body problem. Arch. Ration. Mech. Anal. 194 (2009), 283–313.CrossRefGoogle Scholar
[15]Marchal, C.. How the minimisation of action avoids singularities. Celestial Mech. 83 (2002), 325–354.CrossRefGoogle Scholar
[16]Wintner, A.. The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton, NJ, 1941.Google Scholar