Abstract:
Noether’s theorems for a fractional generalized Birkhoffian system in terms of Caputo derivatives are studied. Firstly, the generalized Pfaff–Birkhoff principle based on Caputo fractional derivatives is established, the fractional generalized Birkhoffian equations are derived. Then, the fractional Noether symmetry and the fractional conserved quantity under special infinitesimal transformations without transforming the time are studied. Noether’s theorem for the fractional generalized Birkhoffian system is established. Once more, the fractional Noether symmetry and the fractional conserved quantity under general infinitesimal transformations with transforming the time are studied, Noether’s theorem for the fractional generalized Birkhoffian system is established. The proof is given by using the time reparametric method. Finally, an example is given to illustrate its application.