# From Classical Trajectories to Quantum Commutation Relations

## TL;DR

This paper explores the geometric structures underlying classical and quantum descriptions of dynamical systems, highlighting ambiguities in reconstructing equations from experimental data and their implications for quantization.

## Contribution

It introduces geometric structures in Lagrangian, Hamiltonian, and quantum frameworks, emphasizing their dependence on observed trajectories and the ambiguities involved.

## Key findings

- Geometric structures are not uniquely determined by observed trajectories.
- Ambiguities in reconstructing differential equations affect quantization.
- The work highlights the role of geometric insights in classical and quantum dynamics.

## 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.