The paper presents a revisitation of the Legendre transformation within a gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold. In this formulation the Lagrangian L is replaced by a section l of a suitable principal fiber bundle over the velocity space, called the lagrangian bundle, while the associated Poincar´e-Cartan 2-form is recognized as the curvature 2-form of a connection induced by l. A parallel construction leads to the identification of a hamiltonian and a co-hamiltonian bundle over the phase space.
A geometric approach to the Legendre transformation
Pagani, Enrico
2004-01-01
Abstract
The paper presents a revisitation of the Legendre transformation within a gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold. In this formulation the Lagrangian L is replaced by a section l of a suitable principal fiber bundle over the velocity space, called the lagrangian bundle, while the associated Poincar´e-Cartan 2-form is recognized as the curvature 2-form of a connection induced by l. A parallel construction leads to the identification of a hamiltonian and a co-hamiltonian bundle over the phase space.File in questo prodotto:
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