A note on the problem of geodesics curves with fractional derivative of Atangana–Baleanu

This paper presents some properties and relations that exist between fractional operators in the sense of Riemann–Liouville and Atangana–Baleanu. In particular, a proof of the integration by parts formula is presented when the fractional derivative of Atangana–Baleanu is considered. As an application of these properties, the classical problem of determining geodesic curves in the plane is analyzed considering the fractional derivative of Atangana–Baleanu. The introduction of the fractional derivative in the functional describing the optimization problem is performed by means of the fractionalization method. The obtained results are compared with the classical problem.