Dynamics determines geometry

TL;DR

This paper explores how the structure of modified Poisson brackets influences the inverse calculus of variations and s-equivalence, highlighting potential discrepancies between classical and quantum equivalence using non-commutative geometry insights.

Contribution

It introduces a novel analysis of s-equivalence and inverse variational problems through non-commutative geometry, emphasizing quantum-level differences.

Findings

Classical s-equivalent systems may not be quantum-mechanically equivalent.

Modified Poisson brackets play a crucial role in system equivalence.

Different approaches to the Nair-Polychronakos oscillator reveal quantum discrepancies.

Abstract

The inverse problem of calculus of variations and $s$-equivalence are re-examined by using results obtained from non-commutative geometry ideas. The role played by the structure of the modified Poisson brackets is discussed in a general context and it is argued that classical $s$-equivalent systems may be non-equivalent at the quantum mechanical level. This last fact is explicitly discussed comparing different approaches to deal with the Nair-Polychronakos oscillator.