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.
Massa, E., Pagani, E., Lorenzoni, P. (2000). On the gauge structure of classical mechanics. TRANSPORT THEORY AND STATISTICAL PHYSICS, 29(1-2), 69-91 [10.1080/00411450008205861].
On the gauge structure of classical mechanics
LORENZONI, PAOLO
2000
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.
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