2004/11/16 by Eduardo Martı́nez, Eduardo Martinez, Martinez, Eduardo
Mathematics · Physics and Astronomy · #35F20 #49S05 #53C15 #53D99 #58A20 #58H99 #70S05 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.DG #math.MP #msc:35F20 #msc:49S05 #msc:53C15 #msc:53D99 #msc:58A20 #msc:58H99 #msc:70S05
paper · pdf · doi:10.48550/arxiv.math/0411352
43 pages
arxiv created 2004/11/16 · openalex publication_date 2004/11/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle we give a variational description of the equations of motion. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincare and Lagrange Poincare cases), variational problems for holomorphic maps, Sigma models or Chern-Simons theories. One of the advantages of our theory is that it is based in the existence of a multisymplectic form on a Lie algebroid.