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On the gauge structure of classical mechanics. (English) Zbl 0968.70014
A well-known property of Lagrangian mechanics is the invariance of Euler-Lagrange equations for a given Lagrangian $$L=L(t,q, \dot q)$$ with respect to gauge transformations $$L\rightarrow L+df/dt$$, $$f=f(t,q)$$. In this paper, in order to give a geometrical interpretation of this invariance, some fiber bundles are introduced: (i) one, called the bundle of affine scalars over the configuration space, (ii) two, called Lagrangian and co-Lagrangian bundles over the velocity space, (iii) two, called Hamiltonian and co-Hamiltonian bundles over the phase space. In this framework, the Lagrangian $$L$$ is replaced by a section of the Lagrangian bundle, while the associated Cartan 2-form is the curvature 2-form of a connection induced by $$L$$ on the co-Lagrangian bundle. A parallel construction is produced for the Hamiltonian formalism.

MSC:
 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics 70H03 Lagrange’s equations 70H05 Hamilton’s equations 70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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