N-Symplectic Analysis of Field Theory

Abstract

Two techniques for relating n-symplectic geometry tothe jet bundle formulations of classical fieldtheory are presented.The tangent bundle of the frame bundle of a manifoldM is shown to be a principal fiber bundle over thejet bundle of the tangent bundle of M. We are ableto generalize this result to symmetric andantisymmetric tensor bundles of rank p. Using thisGL(m) gauge freedom, we interpret the standard freefield Lagrangian as a symmetric type (0,2) tensor onLM. The adapted frame bundle of an arbitrary fiberbundle PI is shown to be a principal bundle over thejet bundle of PI. Using this GL(m) times GL(k) gaugefreedom we generate a modified m+k-symplecticgeometry from a lifted Lagrangian. The modifiedsoldering form is shown to induce the Cartan-Hamilton-Poincare m-form on the jet bundleof PI. We derive generalized Hamilton-Jacobi andHamilton equations on adapted frame bundle, andshow that the Hamilton-Jacobi and canonicalequations of Caratheodory-Rund and de Donder-Weylare obtained as special cases.These results demonstrate that by introducingadditional gauge freedom into the standard jetbundle formalism one can obtain a great of ofadditional geometric and algebraic structure. Suchadditional structure may be key in achieving agreater understanding of field theory or in attempts at quantization.