Fractional lie series and transforms as canonical mappings Dr. Abd El-Salam

Using the Riemann-Liouville fractional differintegral operator, the Lie theory is reformulated. The fractional Poisson bracket over the fractional phase space as 3N state vector is defined to be the fractional Lie derivative. Its properties are outlined and proved. A theorem for the canonicity of the transformation using the exponential operator is proved. The conservation of its generator is proved in a corollary. A Theorem for the inverse fractional canonical mapping is proved. The composite mappings of two successive transformations is defined. The fractional Lie operator and its properties are introduced. Some useful lemmas on this operator are proved. Lie transform depending on a parameter over the fractional phase space is presented and its relations are proved. Two theorems that proved the transformation  = EW Z is completely canonical and is a solution of the Hamiltonian system (30) are given. Recurrence relations are obtained.