Application Of Complex Clifford Algebra To Symplectic Hamlitonian structures

This thesis is an attempt to formulate the symplectic Hamiltonianmechanics and its Birkhoffian generalization using complex Cliffordalgebra. Kahlerian Poisson brackets and Kahlerian Lagrange bracketsare introduced as a generalization of both standard Poisson and Lagrangebrackets. Birkhoffs general relation between the generalized Lagrangianand Birkhoffian is derived and is shown to be the most generalizationbetween them. Two sets of conjugate variables are defined in terms ofthe generalized phase space metric.A comparative study of two papers about the formulation ofHamiltonian mechanics using geometric calculus is done for discussingthe up to date researches in the same field. The study shows that -in fact-the two papers are equivalent in many aspects.Hamiltonian systems with constraints are studied. The Diracbracket, which is used instead of Poisson bracket in the case ofconstrained systems, is derived as a special case of generalized Poissonbracket using matrix algebra. The transformations generated by theDirac bracket is presented and shown to be the normal canonicaltransformations in another set of canonical variables.,