On the gauge structure of classical mechanics

Abstract

A self consistent gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold is proposed. In the resulting set-up, the “Lagrangian” L is replaced by a section of a suitable principal fiber bundle over the velocity space, called the lagrangian bundle, while the associated Poincaré-Cartan 2-form is recognized as the curvature 2-form of a connection induced by L on a second “co-lagrangian” principal bundle. A parallel construction leads to the identification of a hamiltonian and a co-hamiltonian bundle over the phase space. An analysis of the properties of these spaces provides an intrinsic geometrical characterization of the Legendre transformation, thus allowing a systematic translation of the hamiltonian formalism into the newer scheme.

PACS:

• 03.20+1
• 02.40.+m

1991 Mathematical subject classification:

• 70D10
• 70F25

Key Words:

• Lagrangian Dynamics
• Gauge Theories
• Connections