Inverse problem for Lagrangian systems on Lie algebroids and applications to reduction by symmetries Articles uri icon

authors

  • BARBERO LIÑAN, MARIA
  • FARRE PUGGALI, MARTA
  • MARTIN DE DIEGO, DAVID

publication date

  • August 2016

start page

  • 665

end page

  • 691

issue

  • 4

volume

  • 180

International Standard Serial Number (ISSN)

  • 0026-9255

Electronic International Standard Serial Number (EISSN)

  • 1436-5081

abstract

  • The language of Lagrangian submanifolds is used to extend a geometric characterization of the inverse problem of the calculus of variations on tangent bundles to regular Lie algebroids. Since not all closed sections are locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are necessary but not sufficient, so they give a weaker definition of the inverse problem. As an application the Helmholtz conditions on Atiyah algebroids are obtained so that the relationship between the inverse problem and the reduced inverse problem by symmetries can be described. Some examples and comparison with previous approaches in the literature are provided.

keywords

  • lie algebroids; sode section; inverse problem; atiyah algebroid; calculus; geometry; dynamics