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.

Sundman transformation and alternative tangent structures

“…These generalised Sundman transformations have also been used to solve many interesting problems in classical mechanics (see e.g. [9][10][11][12]) and celestial mechanics [13][14][15][16][17].…”

“…Fortunately, the existence of alternative tangent structures will be useful to solve this problem (see e.g. [17]).…”