Research Thesis on Formulation of HAMILTONIAN MECHANICS reconciliation of Classical Mechanics (Langrangian) with Quantum Mechanics (Hamiltonian), smaller infinitesimal particles Einstein's Mechanics. Addendum, Canonical Transformation, Principle of Virtual Work, Harmonic Oscillator and Lemma on Mathematical Method ascertained by PROF. J.C AMAZIGO DEPARTMENT OF MATHEMATICS UNIVERSITY OF NIGERIA NSUKKA.