Homogeneous fractional embeddings

TL;DR

This paper addresses the issue of temporal homogeneity in fractional embeddings of Lagrangian systems by introducing an extrinsic fractional constant, ensuring consistent fractional equations and illustrating the approach with the harmonic oscillator.

Contribution

It introduces the concept of a fractional constant to maintain homogeneity in fractional embeddings of Lagrangian systems, providing a novel formalism.

Findings

Homogeneity can be preserved using an extrinsic fractional constant.

Two equivalent fractional embeddings are constructed.

The approach is exemplified with the harmonic oscillator.

Abstract

Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retain homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.