Geometric Structures in Field Theory
arXiv:math-ph/0208036
Abstract
This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper reviews, compares and constrasts the various generalizations in order to bring some unity to the field of study. The generalizations seem to fall into two categories. In one direction some have generalized the geometric structures of the bundles, arriving at the various axiomatic systems such as k-symplectic and k-tangent structures. The other direction was to fundamentally extend the bundles themselves and to then explore the natural geometry of the extensions. This latter direction gives us the multisymplectic geometry on jet and cojet bundles and n-symplectic geometry on frame bundles.
References in corpus (5)
- Geometry of multisymplectic Hamiltonian first-order field theories
- On Field Theoretic Generalizations of a Poisson Algebra
- Covariant Hamiltonian field theory
- Novel algebraic structures from the polysymplectic form in field theory
- On a Multisymplectic Formulation of the Classical BRST Symmetry for First Order Field Theories Part II: Geometric Structures
Cited by in corpus (16)
- The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory
- Symmetries in Classical Field Theory
- Precanonical quantization of Yang-Mills fields and the functional Schroedinger representation
- Geometric aspects of nonholonomic field theories
- Hamilton-Jacobi Theory in k-Symplectic Field Theories
- Tulczyjew's triples and lagrangian submanifolds in classical field theories
- Hamilton-Jacobi theory in k-cosymplectic field theories
- A class of nonholonomic kinematic constraints in elasticity
- k-cosymplectic classical field theories: Tulczyjew, Skinner--Rusk and Lie-algebroid formulations
- The momentum map for nonholonomic field theories with symmetry
- Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry
- Symmetry aspects of nonholonomic field theories
- Nonlinear Connections on Gerbes, Clifford-Finsler Modules, and the Index Theorems
- Clifford Algebraic approach to the De Donder-Weyl Hamiltonian theory
- n-symplectic quantization on LRn
- Lagrangian submanifolds in k-symplectic settings