### Graduate Course, Spring 2013

7,5 ECTS or 5 ECTS

**Short description:**

The theme of the course is the interaction between geometry, classical mechanics, and numerics. We study continuous dynamical systems from a differential geometric view-point. Particular attention is paid to Lagrangian and Hamiltonian systems, reversible systems, and systems with Lie group symmetries. These types of systems occur in a wide variety of applications, for example, accelerator physics, biology, celestial mechanics, fluid mechanics, molecular dynamics, non-linear control theory, and quantum mechanics. The geometrical insights yield qualitative information about the dynamics. For example, we discuss the Kolmogorov-Arnold-Moser (KAM) theorem on stability of invariant tori for finite dimensional Hamiltonian systems. Another example is Vladimir Arnold’s description of a perfect fluid as a geodesic equation on the group of diffeomorphisms.

A main part of the course deals with construction and analysis of numerical time-stepping algorithms that preserve underlying geometric structures. This branch of computational mathematics, called geometric integration, provides methods with superior qualitative properties, e.g., in terms of conservation of energy, preservation of integrability, and conservation of phase volume. The principal example at hand is symplectic integrators for Hamiltonian systems. We discuss numerical error analysis and its relation to perturbation theory and structural stability of dynamical systems. Throughout the course we highlight concrete examples to illustrate various phenomena.

Parts of the course will be given by the guest teachers Robert McLachlan from Massey University in New Zealand and Olivier Verdier from the University of Bergen in Norway. Both Robert and Olivier are international experts in the field of geometric integration.

**Location and dates:**

Department of Mathematical Sciences, Spring semester 2013.

The course will be lectured daily during three intensive periods:

March 18-22, April 15-19 and May 20-24.

Introductory lecture: Monday March 18, 15.15-17.00, MV:L14.

Lectures during March 19-25:

Tuesday, 08.00-09.45, MV:L14

Wednesday, 08.00-09.45, MV:L14

Thursday, 08.00-09.45, MV:L14

Friday, 08.00-09.45, MV:L14

EXTRA LECTURE: Monday, 15.15-17.00, MV:L15

Lectures during April 15-19:

Monday, 15.15-17.00, MV:L14

Tuesday, 10.00-11.45, MV:L14

Wednesday, 08.00-09.45, MV:L14

Thursday, 15.15-17.15, MV:L11 (notice the room!)

Friday, 08.00-09.45, MV:L14

Lectures during May 20-24:

Monday, 15.15-17.00, MV:L14

Tuesday, 15.15-17.00, MV:L11 (notice the room!) Lab 1, Lab 2

Wednesday, 08.00-09.45, MV:L14

Thursday, 08.00-09.45, MV:L14

Friday, 08.00-09.45, MV:L14

**Aim of the course:**

After finishing the course the student will have a working knowledge of the differential geometric description of classical mechanics, the concept of Lie group symmetries, and on symplectic numerical integrators for Hamiltonian systems. Furthermore, the student will have gained understanding of Liouville-Arnold integrability, KAM-stability, the geometric description of fluid mechanics, and backward error analysis of symplectic integrators.

**Target group:**

Graduate and master students in mathematics, mechanics, physics and automatic control.

**Entry requirements:** Some knowledge of differential geometry.

**Course organizers:** Klas Modin and Stig Larsson

**Teachers:** Robert McLachlan, Klas Modin and Olivier Verdier.

**Lectures:** 14 double hours.

**Examination:** Hand-in problems (5 ECTS) + individual project (7.5 ECTS)

- Hand-in problems week 1: at least 3 exercises from the lecture notes
- Hand-in problems week 2: at least 1 of the programming exercises (UPDATED 2013-04-19)

**Literature:**

The slides from the introductory lecture are available here.

Lecture notes for some of the lectures in the course are available here:

Lecture 18/3, 19/3 (UPDATED 2013-03-19)

Lecture 20/3, 21/3

Lecture 22/3, 25/3

Lecture 22/3, 25/3

Lecture 23/5, 24/5

Survey articles on geometric integration (course literature for GI lectures):

- McLachlan and Quispel, Geometric Integrators for ODEs, J. Phys. A (2006)
- Hairer, Lubich, Wanner, Geometric Numerical Integration Illustrated by the Störmer/Verlet Method, Acta Numerica (2003)
- Marsden and West, Discrete Mechanics and Variational Integrators, Acta Numerica (2001)

As side literature on differential geometry and geometric mechanics, I recommend the following books:

- Mathematical Methods of Classical Mechanics, by V. Arnold

Chapters 3,4,6,7,8 - Introduction to Mechanics and Symmetry, by J. E. Marsden and T. S. Ratiu

Chapters 4,7,2,5,9

There is also a Primer on Geometric Mechanics by Christian Lessig available online.

**Registration:** Please email the course organizers for registration.

**Examples of simulations using geometric integrators (videos):**

9 billion molecules simulation

Water phase changes