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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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References:

[1] Bleecker D., Gauge Theory and Variational Principles (1981) · Zbl 0481.58002
[2] Massa E., Ann. Inst. Henri Poincaré, Physique théorique 55 pp 511– (1991)
[3] Massa E., Ann. Inst. Henri Poincaré, Physique théorique 61 pp 17– (1994)
[4] Massa E., Ann. Inst. Henri Poincaré, Physique théorique 66 pp 1– (1997)
[5] DOI: 10.1063/1.525252 · Zbl 0507.70022 · doi:10.1063/1.525252
[6] DOI: 10.1016/0003-4916(82)90334-7 · Zbl 0501.70020 · doi:10.1016/0003-4916(82)90334-7
[7] Kobayashi Y., Foundations of Differential Geometry (1969) · Zbl 0175.48504
[8] Cantrijn F., Reduction of degenerate Lagrangian systems. JGP 3 pp 353– (1986) · Zbl 0621.58020
[9] Garcia P., J. Differential Geometry 12 pp 351– (1977)
[10] Giachetta G., New Lagrangian and Hamiltonian Methods in Field Theory (1997) · Zbl 0913.58001
[11] Kolář I., Natural Operations in Differential Geometry (1993) · Zbl 0782.53013
[12] Abraham R., Foundations of Mechanics (1978)
[13] Arnold V. I., Mathematical Methods of Classical Mechanics (1975)
[14] Godbillon C., Géométrie Différentielle et Mécanique Analytique (1969)
[15] Saunders D. J., Lecture Note Series 142, in: London Mathematical Society (1989)
[16] DOI: 10.1088/0305-4470/17/7/011 · Zbl 0545.58020 · doi:10.1088/0305-4470/17/7/011
[17] DOI: 10.1088/0305-4470/17/10/012 · Zbl 0542.58011 · doi:10.1088/0305-4470/17/10/012
[18] de Leon M., Methods of Differential Geometry in Analytical Mechanics (1989)
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