A fractional product Pα[f,g] whose derivative is exactly the Leibniz-type quantity Dαa+[f] · g − f · Dαb−[g]. Multiple applications and perspectives, including fractional Noether theorems and constants of motion for fractional Hamiltonian systems.
Abstract
The present paper aims at providing a significant contribution to a challenge opened in 2007 in the field of fractional calculus of variations. Precisely, in an effort to obtain a fractional version of Noether’s theorem, several authors have found themselves faced with a Leibniz-type equality of the form Dαa+[f] · g − f · Dαb−[g] = 0ℝn. Hence, to deduce a relevant constant of motion, the challenge is, not only to identify an expression whose derivative is exactly the above Leibniz-type quantity, but a practical expression that does, in the sense that it has to be sufficiently compact and structured to be amenable to analytical investigations. This is exactly the content of the present work that introduces and studies the so-called fractional product Pα[f,g] (which corresponds moreover to a fractional extension of the classical product). In addition to fractional Noether theorems, it turns out that the fractional product Pα[f,g] is also valuable to establish two additional new results in the field of fractional calculus of variations: a fractional Beltrami theorem and a constant of motion for fractional Hamiltonian systems. Also, thanks to this new operator, the well-known fractional integration by parts formula is extended, for the first time in the literature, to subintervals.
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BibTeX
@misc{bourdin2026fractional,
title = {{A fractional product $\mathrmP^\alpha[f,g]$ whose derivative is exactly the Leibniz-type quantity $\mathrmD^\alpha_a+[f] \cdot g - f \cdot \mathrmD^\alpha_b-[g]$. Multiple applications and perspectives, including fractional Noether theorems and constants of motion for fractional Hamiltonian systems.}},
author = {Loïc Bourdin},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05773571v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - A fractional product P^α[f,g] whose derivative is exactly the Leibniz-type quantity D^α_a+[f] · g − f · D^α_b−[g]. Multiple applications and perspectives, including fractional Noether theorems and constants of motion for fractional Hamiltonian systems.
AU - Loïc Bourdin
PY - 2026
DA - 2026/10/01
PB - HAL
UR - https://hal.science/hal-05773571v1
AB - The present paper aims at providing a significant contribution to a challenge opened in 2007 in the field of fractional calculus of variations. Precisely, in an effort to obtain a fractional version of Noether’s theorem, several authors have found themselves faced with a Leibniz-type equality of the form D^α_a+[f] · g − f · D^α_b−[g] = 0_ℝ^n. Hence, to deduce a relevant constant of motion, the challenge is, not only to identify an expression whose derivative is exactly the above Leibniz-type quantity, but a practical expression that does, in the sense that it has to be sufficiently compact and structured to be amenable to analytical investigations. This is exactly the content of the present work that introduces and studies the so-called fractional product P^α[f,g] (which corresponds moreover to a fractional extension of the classical product). In addition to fractional Noether theorems, it turns out that the fractional product P^α[f,g] is also valuable to establish two additional new results in the field of fractional calculus of variations: a fractional Beltrami theorem and a constant of motion for fractional Hamiltonian systems. Also, thanks to this new operator, the well-known fractional integration by parts formula is extended, for the first time in the literature, to subintervals.
ER -