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

Document Type: Regular Paper



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.