Noether’s theorem for fractional Birkhoffian systems of variable order

Abstract

The Noether symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order are studied. Firstly, the Pfaff–Birkhoff–d’Alembert principle within fractional derivatives of variable order is obtained, and corresponding variable order fractional Birkhoff’s equations are deduced. Secondly, the invariance of the fractional Pfaff action of variable order is studied under the one-parameter group of infinitesimal transformations, and the definition of the variable order fractional conserved quantity is given. Finally, the Noether’s theorem for the fractional Birkhoffian system of variable order is established. At the end of this paper, an example is given to illustrate the application of the results.