Math 547 - Symplectic Geometry and Mechanics

Spring 2013

Section information: Classes meet on Tuesdays and Thursdays from 1:00 PM to 2:30 PM at Cupples I 216.

Subject: This is a course on classical mechanics and symplectic geometry. It is mainly aimed at graduate students of mathematics, but other students are welcome. The emphasis is on the geometric underpinnings of classical classical mechanics of holonomic and non-holonomic systems, emphasizing physical examples. No specific knowledge of physics is assumed, but familiarity with manifold theory at the level of Math 5041 is strongly recommedned.

Text: I plan to follow in part the text: Nonholonomic Mechanics and Control by A.M. Bloch
Springer, Interdisciplinary Appled Mathematics series, V. 24, 2003.

I may also distribute occasional course notes.

Tentative list of topics: Here's what I expect we'll be able to cover during the semester, not necessarily in this order.

• Differential geometry (distributed along the course):
  • Manifolds, Lie groups, vector and tensor fields, flows;
  • Vector and principal bundles, connections, curvature;
  • Symplectic and Poisson manifolds, and Hamiltonian flows.
  • Basics of Riemannian manifolds.
• Assorted examples of mechanical systems:
  • Rigid bodies;
  • Linked rigid bodies;
  • Roling discs and balls;
  • Chaplygin sleigh, the planar skater, and various other non-holonomic systems;
  • Collisions.
• Theory:
• Lagrangian mechanics and variational principles;
• Lie-Poisson brackets, the Euler-Poincare equations;
• Momentum maps;
• The Lagrange-d'Alembert principle;
• Systems with symmetry;
• Etc.

Coursework: Coursework will involve solving assigned problems and/or giving a presentation in class at some point during the semester. More details about what is expected for an A will be discussed in class.