ME726A - Hamiltonian Mechanics and Sympletic Algorithms

Part 1a
Some relevant definitions and results in the theory of differentiable manifolds, smooth vector fields, differential forms, exterior) calculus (differentiation and integration using differential forms),differential equatio11s and their associated flow maps, Symplectic manifolds.

Part 1b
Brief review of Hamiltonian mechanics (Lagrange's vs Hamilton's Equations) , Canonical Transformation , Legendre Traus formation, Symplectic Iansfonnations, Some definitions and: results in the theory of Continuous Groups for Symmetries and Conserved quantities, Poincare Cartan invariant , The Hamilton Jacobi Partial Differential Equation. Integrable systems (simple examples).

Part 2a
Some basic notions of numerical algorithms (order conditions etc). Examples of Numerical methods, Symplectic Integra tors, and Geometric integ Tators. Applications to simple problems in particle dynamics and a two body problem.

Part 2b
Symplectic Runge Kutta lV methods, Generating Function for Symplectic Rm1geKutta Methods and Symplectic I Methods Based on it. Variational Integrators. Introduction to Hamiltonian Perturbation theory (if time permits). Discussion on some open problems in symplectic algorithms and a brief discussion on geometric numerical integration with some applications to mechanical systems.