From Lagrangian to Quantum Mechanics with Symmetries

TL;DR

This paper revisits Jacobi's classical method for deriving Lagrangians using Lie symmetries, demonstrating how it facilitates the transition from classical to quantum mechanics through Noether's theorem and symmetry preservation.

Contribution

It introduces a forgotten Jacobi method for finding Lagrangians and links symmetry preservation to quantization, providing a novel perspective on classical-quantum transition.

Findings

Jacobi's method can generate multiple Lagrangians for simple models.

Lie symmetries lead to Jacobi last multipliers and Lagrangians.

Preserving Noether symmetries in Schrödinger equations aids quantization.

Abstract

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.