Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem

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Abstract
It is shown that the study of nonbijective transformation requires a fiber bundle formulation of mechanics. The conditions upon which nonbijective canonical point transformations can be defined are given. Then, as an example, we apply that theory to the study of the Coulomb problem in two and three dimensions. The Hopf fibration leads to an inverse harmonic oscillator problem. Since the completion of this work, a paper by G. H. Ringwood and J. T. Devreese has been published in J. Math. Phys. 21, 1390 (1980), dealing with the same problem. Their work is based on the construction of propagators in quotient spaces. The identity between the propagator prescriptions and nonlinear canonical transformations is not automatically fulfilled. Therefore it seems that the reliability of their results is not due to their method, which in general is not correct, but to an underlying property of the transformation used, namely the Kustaanheimo-Stiefel map (see our results).