Variational procedure for higher-derivative mechanical models in a fractional integral

¹ Departamento de Física, Universidade Federal Rural do Rio de Janeiro (UFRRJ) - BR 465-07, 23890-971, Seropédica, RJ, Brazil
² Department of Physics, Centro Federal de Educação Tecnológica Celso Suckow da Fonseca Av Maracanã 229, 20271-110, Rio de Janeiro, RJ, Brazil
³ Universidade Federal Rural do Rio de Janeiro, UFRRJ- IM/DTL - Av. Governador Roberto Silveira s/n, Nova Iguacu, Rio de Janeiro, RJ, Brazil
⁴ Centro Brasileiro de Pesquisas Físicas (CBPF) - Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, RJ, Brazil

^(a) crgodinho@gmail.com
^(b) nelsonpanza@gmail.com
^(c) josewebe@gmail.com
^(d) helayel@cbpf.br

Received: 31 July 2019
Accepted: 30 March 2020

Abstract

We present both the Lagrangian and Hamiltonian procedures to treat higher-derivative equations of motion for mechanical models by adopting the Riemann-Liouville fractional integral to formulate their respective actions. Our focus is the possible interplay between fractionality and a dynamics based on higher derivatives. We point out and discuss the efficacy and difficulties of this approach. We also contemplate physical and geometrical interpretations and present details of the inspection we carry out by considering an explicit situation, that of a higher-derivative harmonic oscillator. Additionally, we have also used a recent proposal of a variational approach with local deformed derivatives. In this context, we have derived a complete set of linear and non-linear equations for a Pais-Uhlenbeck–type oscillator.