Further Research for Lagrangian Mechanics within Generalized Fractional Operators

Abstract

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

“…Atanacković et al [9] derived the fractional Noether theorem under the classical definition of conserved quantity, which reveals the inherent connection between Noether symmetry transformations and fractional-order conserved quantities. In recent years, the study of conserved quantities and symmetries in fractional mechanics using variational methods has made some headway [10][11][12][13][14][15][16][17][18].…”