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Proceedings of the American Mathematical Society
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
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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$).


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Additional Information

Ezequiel Maderna
Affiliation: CMAT, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
Email: emaderna@cmat.edu.uy

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11542-X
PII: S 0002-9939(2013)11542-X
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.