Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Translation invariance of weak KAM solutions of the Newtonian $ N$-body problem

Author: Ezequiel Maderna
Journal: Proc. Amer. Math. Soc. 141 (2013), 2809-2816
MSC (2010): Primary 37J15, 70H20, 49L25
Published electronically: April 19, 2013
MathSciNet review: 3056571
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Hamilton-Jacobi equation $ H(x,d_xu)=c$, where $ c\geq 0$, of the classical $ N$-body problem in some Euclidean space $ E$ of dimension at least two. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant ($ c=0$), also called weak KAM solutions. We show that all these solutions are invariant under the action by translations of $ E$ in the space of configurations. We also show the existence of non-invariant solutions for the supercritical equations ($ c>0$).

References [Enhancements On Off] (What's this?)

  • 1. A. Da Luz and E. Maderna, On the free time minimizers of the Newtonian $ N$-body problem, preprint,
  • 2. A. Chenciner, Action minimizing solutions of the Newtonian $ n$-body problem: From homology to symmetry, Proceedings of the ICM, Vol. III (Beijing, 2002), 279-294, Higher Ed. Press, Beijing, 2002. MR 1957539 (2004f:70026a)
  • 3. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 690039 (85g:35029)
  • 4. D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $ n$-body problem, Invent. Math. 155 (2004), no. 2, 305-362. MR 2031430 (2005b:70010)
  • 5. E. Maderna, Invariance of global solutions of the Hamilton-Jacobi equation, Bull. Soc. Math. France 130 (2002) no. 4, 493-506. MR 1947450 (2004b:37132)
  • 6. E. Maderna, On weak KAM theory for $ N$-body problems, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 1019-1041. MR 2995654

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37J15, 70H20, 49L25

Retrieve articles in all journals with MSC (2010): 37J15, 70H20, 49L25

Additional Information

Ezequiel Maderna
Affiliation: CMAT, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay

Received by editor(s): May 30, 2011
Received by editor(s) in revised form: November 7, 2011
Published electronically: April 19, 2013
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society