Invariance of global solutions of the Hamilton-Jacobi equation

Ezequiel Maderna

Bulletin de la Société Mathématique de France (2002)

  • Volume: 130, Issue: 4, page 493-506
  • ISSN: 0037-9484

Abstract

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We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.

How to cite

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Maderna, Ezequiel. "Invariance of global solutions of the Hamilton-Jacobi equation." Bulletin de la Société Mathématique de France 130.4 (2002): 493-506. <http://eudml.org/doc/272418>.

@article{Maderna2002,
abstract = {We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.},
author = {Maderna, Ezequiel},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Hamilton-Jacobi; lagrangian; symmetries},
language = {eng},
number = {4},
pages = {493-506},
publisher = {Société mathématique de France},
title = {Invariance of global solutions of the Hamilton-Jacobi equation},
url = {http://eudml.org/doc/272418},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Maderna, Ezequiel
TI - Invariance of global solutions of the Hamilton-Jacobi equation
JO - Bulletin de la Société Mathématique de France
PY - 2002
PB - Société mathématique de France
VL - 130
IS - 4
SP - 493
EP - 506
AB - We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.
LA - eng
KW - Hamilton-Jacobi; lagrangian; symmetries
UR - http://eudml.org/doc/272418
ER -

References

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  1. [1] G. Contreras, J. Delgado & R. Iturriaga – « Lagrangian flows: the dynamics of globally minimizing orbits, II », Bol. Soc. Bras. Mat. 28 (1997), no. 2, p. 155–196. Zbl0892.58065MR1479500
  2. [2] A. Fathi – « Solutions KAM faibles conjugués et barrières de Peierls », C. R. Acad. Sci. Paris, Série I 325 (1997), p. 649–652. Zbl0943.37031MR1473840
  3. [3] —, « Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens », C. R. Acad. Sci. Paris, Série I 324 (1997), p. 1043–1046. Zbl0885.58022MR1451248
  4. [4] —, « Weak KAM Theorem in Lagrangian Dynamics », Preprint, 2000. 
  5. [5] A. Fathi & E. Maderna – « Weak KAM Theorem on Non Compact Manifolds », Preprint, 2000. Zbl1139.49027MR2346451
  6. [6] S. Kobayashi – Transformation Groups in Differential Geometry, Springer-Verlag, 1995, reprint of the 1972 ed. Zbl0246.53031MR355886
  7. [7] R. Mañé – « Lagrangian flows: the dynamics of globally minimizing orbits », Bol. Soc. Bras. Mat. 28 (1997), no. 2, p. 141–153. Zbl0892.58064MR1479499
  8. [8] J. Mather – « Action minimizing measures for positive definite Lagrangian systems », Math. Z.207 (1991), p. 169–207. Zbl0696.58027MR1109661
  9. [9] D. Montgomery & L. Zippin – Transformation Groups, Interscience Tracts, vol. 1, J. Wiley & Sons, 1955. MR73104
  10. [10] G. Paternain & M. Paternain – « Critical values of autonomous Lagrangian systems », Comment. Math. Helvetici72 (1997), p. 481–499. Zbl0921.58017MR1476061
  11. [11] W. Ziemer – Weakly Differentiable Functions, Springer-Verlag, 1989. Zbl0692.46022MR1014685

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