Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space

This paper focuses on studying Noether's theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by El-Nabulsi. Both holonomic and nonholonomic systems are studied. First, the fractional variational problem from extended exponentially fractional integral, as well as El-Nabulsi-Hamilton's canonical equations are established; second, the definitions and criteria of fractional Noether symmetric transformations and fractional Noether quasi-symmetric transformations are presented which are based on the invariance of El-Nabulsi-Hamilton action under the infinitesimal group transformations; finally, the fractional Noether's theorem is established, which reveals the inner relationship between a fractional Noether symmetry and a fractional conserved quantity.