Translation invariance of weak KAM solutions of the Newtonian $N$-body problem
HTML articles powered by AMS MathViewer
- by Ezequiel Maderna PDF
- Proc. Amer. Math. Soc. 141 (2013), 2809-2816 Request permission
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
- A. Da Luz and E. Maderna, On the free time minimizers of the Newtonian $N$-body problem, preprint, http://premat.fing.edu.uy/2009.htm
- A. Chenciner, Action minimizing solutions of the Newtonian $n$-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 279–294. MR 1957539
- Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
- Davide L. Ferrario and Susanna Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math. 155 (2004), no. 2, 305–362. MR 2031430, DOI 10.1007/s00222-003-0322-7
- Ezequiel Maderna, Invariance of global solutions of the Hamilton-Jacobi equation, Bull. Soc. Math. France 130 (2002), no. 4, 493–506 (English, with English and French summaries). MR 1947450, DOI 10.24033/bsmf.2427
- Ezequiel Maderna, On weak KAM theory for $N$-body problems, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 1019–1041. MR 2995654, DOI 10.1017/S0143385711000046
Additional Information
- Ezequiel Maderna
- Affiliation: CMAT, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
- Email: emaderna@cmat.edu.uy
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2809-2816
- MSC (2010): Primary 37J15, 70H20, 49L25
- DOI: https://doi.org/10.1090/S0002-9939-2013-11542-X
- MathSciNet review: 3056571