ON THE GAUGE STRUCTURE OF CLASSICAL MECHANICS

ON THE GAUGE STRUCTURE OF CLASSICAL MECHANICS ∗ Enrico Massa Dipartimento di Matematica dell’Universit`a di Genova Via Dodecaneso, 35 - 16146 Genova (Italia) E-mail: [email protected] Enrico Pagani Dipartimento di Matematica dell’Universit`a di Trento Via Sommarive, 14 - 38050 Povo di Trento (Italia) E-mail: [email protected] Paolo Lorenzoni Via Cesarea, 10 - 16121 Genova (Italia)

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 Poincar´e-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 Keywords: Lagrangian Dynamics, Gauge Theories, Connections.

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This research was partly supported by the National Group for Mathematical Physics of the Italian Research Council (CNR), and by the Italian Ministry for Public Education, through the research project “Metodi Geometrici e Probabilistici in Fisica Matematica”.

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Introduction

A well known feature of Classical Mechanics is the invariance of Lagrange’s equations ∂L d ∂L − k =0 k dt ∂ q˙ ∂q under arbitrary transformations of the form L 7→ L0 := L + df /dt , f = f (t, q) being any smooth function over the configuration manifold, and df /dt denoting the symbolic time derivative of f . In this respect, rather than expressing a mechanical attribute of the system, the Lagrangian L has the nature of a “potential” for the determination of the equations of motion, defined up to a gauge, in the sense commonly employed in classical field theory — e.g. in Classical Electrodynamics [1]. In spite of being entirely obvious, this observation turns out to have interesting implications, providing a deeper insight into the structure of Classical Lagrangian Mechanics. In this paper we present a discussion of this point, both from the lagrangian and from the hamiltonian viewpoint. The gauge structure of the theory is accounted for by introducing a fiber bundle P with structural group