Classical and Quantum Physics pp 163-185 | Cite as

# From Classical Trajectories to Quantum Commutation Relations

## Abstract

In describing a dynamical system, the greatest part of the work for a theoretician is to translate experimental data into differential equations. It is desirable for such differential equations to admit a Lagrangian and/or an Hamiltonian description because of the Noether theorem and because they are the starting point for the quantization. As a matter of fact many ambiguities arise in each step of such a reconstruction which must be solved by the ingenuity of the theoretician. In the present work we describe geometric structures emerging in Lagrangian, Hamiltonian and Quantum description of a dynamical system underlining how many of them are not really fixed only by the trajectories observed by the experimentalist.

## References

- 1.Aristotele.
*Physics. English translation by Philip Wicksteed and Francis Cornford*(Heinemann, London, 1957)Google Scholar - 2.G. Marmo, E.J. Saletan, A. Simoni, B. Vitale,
*Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction*(Wiley-Interscience, New York, 1985)zbMATHGoogle Scholar - 3.A. D’Avanzo, G. Marmo, Reduction and unfolding: the Kepler problem. Int. J. Geom. Methods Mod. Phys.
**2**(1), 83–109 (2005)MathSciNetCrossRefGoogle Scholar - 4.A. D’Avanzo, G. Marmo, A. Valentino, Reduction and unfolding for quantum systems: the Hydrogen atom. Int. J. Geom. Methods Mod. Phys.
**2**(6), 1043–1062 (2005)MathSciNetCrossRefGoogle Scholar - 5.V.I. Man’ko, G. Marmo, E.C.G. Sudarshan, F. Zaccaria, f-oscillators and nonlinear coherent states. Phys. Scr.
**55**(5), 528 (1997)ADSCrossRefGoogle Scholar - 6.L. Schiavone, From trajectories to commutation relations. Master’s thesis, Università degli studi di Napoli Federico II, 2018Google Scholar
- 7.G. Marmo, G. Mendella, W. Tulczyjew, Constrained Hamiltonian system a implicit differential equations. J. Phys. A
**30**, 277–293 (1997)ADSMathSciNetCrossRefGoogle Scholar - 8.D. Krupka,
*Introduction to Global Variational Geometry*(Atlantis Press, Paris, 2015)CrossRefGoogle Scholar - 9.O. Krupkova,
*The Geometry of Ordinary Variational Equations*(Springer, Berlin, 1997)CrossRefGoogle Scholar - 10.G. Giachetta, L. Mangiarotti, J. Sardanashvily,
*New Lagrangian and Hamiltonian Methods in Field Theory*(World Scientific, Singapore, 1996)zbMATHGoogle Scholar - 11.J. Carinena, G. Marmo, A. Ibort, G. Morandi,
*Geometry from Dynamics, Classical and Quantum*(Springer, Heidelberg, 2015)CrossRefGoogle Scholar - 12.R.M. Santilli,
*Foundations of Theoretical Mechanics I: The Inverse problem in Newtonian Mechanics*(Springer, New York, 1978)CrossRefGoogle Scholar - 13.G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo, C. Rubano, The inverse problem in the Calculus of Variations and the geometry of the tangent bundle. Phys. Rep.
**188**(3&4), 147–284 (1990)ADSMathSciNetCrossRefGoogle Scholar - 14.M. Crampin, On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics. J. Phys. A Math. Gen.
**14**, 2567–2575 (1981)ADSMathSciNetCrossRefGoogle Scholar - 15.O. Krupkova,
*Second Order Ordinary Differential Equations on Jet Bundles and the Inverse Problem of the Calculus of Variations. Handbook of Global Analysis*(Elsevier, New York, 1997)Google Scholar - 16.M. Barbero-Linan, M. FarrePuiggalí, D.M. De Diego, Isotropic submanifolds and the inverse problem for mechanical constrained systems. J. Phys. A Math. Theor.
**48**(4), 045210 (2015)ADSMathSciNetCrossRefGoogle Scholar - 17.M. Barbero-Linan, M. Farre Puiggalí, S. Ferraro, D.M. DeDiego, The inverse problem of the calculus of variations for discrete systems. J. Phys. A Math. Theor.
**51**(18), 185202 (2018)ADSMathSciNetCrossRefGoogle Scholar - 18.M. Giordano, G. Marmo, G. Rubano, The inverse problem in the Hamiltonian formalism: integrability of linear Hamiltonian fields. Inverse Prob.
**9**, 443–467 (1993)ADSMathSciNetCrossRefGoogle Scholar - 19.S. De Filippo, G. Landi, G. Marmo, G. Vilasi. Tensor fields defining a tangent bundle structure,
*Annales de l’I.H.P.*,**A**(50-2):205–218 (1989)Google Scholar - 20.R. Abraham, J.E. Marsden, T. Ratiu,
*Manifolds, Tensor Analysis, and Applications*, 3rd edn. (Springer, New York, 2012)zbMATHGoogle Scholar - 21.C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke,
*Poisson Structures*(Springer, Berlin, 2013)CrossRefGoogle Scholar - 22.R.G. Smirnov, Magri–Morosi–Gel’fand–Dorfman’s bi-Hamiltonian constructions in the action-angle variables. J. Math. Phys.
**38**(12), 6444–6453 (1997)ADSMathSciNetCrossRefGoogle Scholar - 23.E. Ercolessi, A. Ibort, G. Marmo, G. Morandi, Alternative linear structures for classical and quantum systems. Int. J. Mod. Phys. A
**22**(18), 3039–3064 (2007)ADSMathSciNetCrossRefGoogle Scholar - 24.E. Ercolessi, G. Marmo, G. Morandi, From equations of motion to canonical commutation relations. Riv. Nuovo Cimento
**33**, 401–590 (2010)Google Scholar - 25.J. Grabowski, M. Kùs, G. Marmo, T. Shulman, Geometry of quantum dynamics in infinite-dimensional Hilbert space. J. Phys. A Math. Theor.
**51**, 165301 (2018)ADSMathSciNetCrossRefGoogle Scholar