Abstract
The aim of this lecture is to give a geometrical formulation to mechanics based on the fibre bundle approach. Here the base manifold is the time and the fibre is either the phase space or its quantized version in case of quantum mechanics.
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See for example, the Hamiltonian (Hamilton-Jacobi) Lagrangian and Poisson formulations of (CM).
Recall, for example, the definitions of classical observables in terms of vector fields with respect to the symplectic form.
Three of these are quantizable but the gravitation is not. This impossibility requires another dynamical conception of space-time which may be introduced within this geometrization programm separately.
For a detailed review see among others: T. Eguchi, et al, Phys. Rep. 66, 1980, 213.
F. Ghaboussi; Proceedings of XXth Conference on “Differential” Geometric Methods in theoretical physics” edited by S. Catto, A. Rocha, 1992, World publishers.
Q is the configuration space. Furthermore, we use here for the simplicity the one-dimensional notation (q ∈ Q).
A true two form on our vector bundle must have dq Λ dq components comparing with dx μ Λ dx ν components for a principal bundle. It is worth mentioning that in Lagrangian formulation of sypmplectic mechanics, similar terms (proportional to dq a Λ dq b, for dimension > 2) appear.
This is similar to the case of standard model SU(2) × U(1). Furthermore, one can also use a “quantized” phase space instead of Q × U(1)-fibre.
All bundle terminologies like “local” or covariancy have to be considered with respect to the time manifold.
We have already shown (Ref 4b) that the Galilein group play the role of gauge group in non-relativistic mechanics. This means that we attach a Galilein frame to the mechanical system under consideration, i.e. the velocity of system is the velocity in Galilein transformation. In this manner we arrive at a covariant constancy with respect to the Galilein transformations.
There is an enormous literature on this topic related with diverse Sagnac type effect. See among others
D. Dieks and G.N: Lenhuis, Am. J. Phys. 58 7, (1990), 650.
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© 1993 Springer Science+Business Media New York
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Ghaboussi, F. (1993). Geometrization of Mechanics. In: Gruber, B. (eds) Symmetries in Science VI. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1219-0_22
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DOI: https://doi.org/10.1007/978-1-4899-1219-0_22
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