This work is intended for graduate students and research mathematicians interested in geometry, topology and analysis.Introduction and some salient features of the model Hamiltonian; Symplectic geometry and the splitting of invariant manifolds; Estimating the splitting matrix using normal forms; The Hamilton-Jacobi method for a simple resonance; Appendix. Invariant tori with vanishing or zero torsion; Bibliography; Introduction and some salient features of the model Hamiltonian; Symplectic geometry and the splitting of invariant manifolds; Estimating the splitting matrix using normal forms; The Hamilton-Jacobi method for a simple resonance; Appendix. Invariant tori with vanishing or zero torsion; Bibliography