ME 726A: HAMILTONIAN MECHANICS AND SYMPLECTIC ALGORITHMS (3-0-0-9)

Some relevant definitions and results in the theory of differentiable manifolds, smooth vector fields, differen- tial forms, (exterior) calculus (differentiation and integration using differential forms), differential equations and their associated flow maps, Symplectic manifolds; Brief review of Hamiltonian mechanics (Lagrange’s vs Hamilton’s Equations), Canonical Transformation, Legendre Transformation, Symplectic Transformations, Some definitions and results in the theory of Continuous Groups for Symmetries and Conserved quantities, Poincar´e-Cartaninvariant, The Hamilton-Jacobi Partial Differential Equation. Integrable systems (sim- ple examples); Some basic notions of numerical algorithms (order conditions etc). Examples of Numerical methods, Symplectic Integrators, and Geometric integrators. Applications to simple problems in particle dynamics and a two body problem; Symplectic Runge-Kutta Methods, Generating Function for Symplec- tic Runge-Kutta Methods and Symplectic Methods Based on it. Variational Integrators. Introduction to Hamiltonian Perturbation theory (if time permits). Discussion on some open problems in symplectic al- gorithms and a brief discussion on geometric numerical integration with some applications to mechanical systems.

Lecture wise Breakup

(I) Lecture 1–2 (2 Lectures): (Part 0: Introduction) 1. Overview of the course. Discussion on some applications, to mechanical systems, of geometric numerical integration. 2. Some applications of symplectic algorithms in understanding dynamics (of conservative systems). Dis- cussion on constrained mechanical systems.

(II) Lecture 3–12 (10 Lectures): (Part 1a: Differential Forms, Vector Fields and Manifolds) 1. Sets, Mappings, Structure of Rn, Differentiation and Integration in Rn, Diffeomorphisms in Rn. 2. Dual Vector Space, Exterior Forms in Rn (1-form, 2-form, k-form). 3. Exterior Product. 4. Differentiable Manifolds, Charts, Tangent Vectors, Vector fields. 5. Tangent Bundle, Cotangent bundle, Differential Forms. 6. Behavior of Differential Forms Under Mappings. 7. Exterior Derivative. 8. Lie Derivative, Lie algebra of vector fields. 9. Chains (1-chain, 2-chain, k-chain), Boundary of Chains, Integration of differential forms, Stokes’ theo- rem. 10. Symplectic Structure in Rn, Symplectic Manifolds. (III) Lecture 13–23 (11 Lectures): (Part 1b: Hamiltonian mechanics) 1. Newtonian dynamics of n interacting particles (potential based interaction). Newton’s, Lagrange’s and Hamilton’s equations for n interacting particles. Legendre Transformation. 2. Configuration space and Phase space. 3. Hamiltonian function and vector fields, Lie algebra of Hamiltonians. 4. Differential equations and their associated flow maps, Phase flows. 5. Canonical Transformation. 6. Symplectic Transformations, Symplectic maps and Hamiltonian flow maps. 7. Invariants. 8. Poincar´e-Cartaninvariant. 9. Some definitions and results in the theory of Continuous Groups for Symmetries and Conserved quan- tities. 10. The Hamilton-Jacobi Partial Differential Equation. 11. Integrable systems (simple examples).

(IV) Lecture 24–30 (7 Lectures): (Part 2a: Examples of Numerical Algorithms) 1. Some basic notions of numerical algorithms (order conditions etc). 2. Examples of Numerical methods. 3. Symplectic Integrators and Geometric integrators. 4. Applications to simple problems in particle dynamics and a two body problem. 5. One-step methods, Numerical example. Higher-order methods, Numerical example. 6. Runge-Kutta methods. 7. Forward vs backward error analysis.

(V) Lecture 31–40 (10 Lectures): (Part 2b: Symplectic Algorithms) 1. Construction of symplectic methods by Hamiltonian splitting. Case studies. 2. Symplectic Runge-Kutta Methods (1). 3. Symplectic Runge-Kutta Methods (2). 4. Generating Function for Symplectic Runge-Kutta Methods (1). 5. Generating Function for Symplectic Runge-Kutta Methods (2). 6. Symplectic Methods Based on Generating Functions. 7. Numerical experiments. 8. Variational Integrators. 9. Introduction to Hamiltonian Perturbation theory (if time permits). 10. Discussion on some open problems in symplectic algorithms. References

1. Arnold, V. I., 1989. Mathematical Methods of Classical Mechanics. Springer. Second edition. [Text- book: for part 1 only sections 18, 32–41, 44–48, for part 2 only sections 13–17, 19]. 2. Leimkuhler, B., Reich, S., 2004. Simulating Hamiltonian Dynamics. Cambridge University Press [Textbook: for part 2 chapters 1, 2, 4–7, 9]. 3. Hairer, E., Lubich, C., Wanner, G., 2006. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer.