2009/06/30 by Cedric M. Campos, Cédric M Campos, Manuel de Leon +4 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Affine transformation #Arbitrariness #Covariant Hamiltonian field theory #Formalism (music) #Hamiltonian (control theory) #Hamiltonian mechanics #Homotopy and Cohomology in Algebraic Topology #Lagrangian #Legendre polynomials #Nonlinear Waves and Solitons #math-ph #math.DG #math.MP #math.SG #msc:53C80 #msc:55R10 #msc:70H50 #msc:70S05
paper · pdf · doi:10.1088/1751-8113/42/47/475207
published as J.Phys.A42:475207,2009 · 21 pages; 4 diagrams; (this version) corrected typos; moved paragraphs; published
openalex publication_date 2009/11/06 · arxiv created 2010/02/05 · arxiv updated 2010/02/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The aim of this paper is to propose an unambiguous intrinsic formalism for higher order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, and implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher order jet bundle and the canonical multisymplectic form on its affine dual. As both of these objects are uniquely defined, the Skinner–Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler–Lagrange equations for higher order field theories. Several examples illustrate our construction.