Graduate course MAT 1126HF

INTRODUCTION TO NONHOLONOMIC MECHANICS AND GEOMETRY
B. Khesin

Nonholonomic mechanics describes the geometry of systems subordinated 
to nonholonomic constraints, i.e systems whose restrictions on velocities 
do not arise from the constraints on the configuration space. 
The best known examples of such systems are a sliding skate and 
a rolling ball, as well as their numerous generalizations.


Syllabus (the numbers below approximately correspond to the week numbers):

I. Nonholonomic mechanics (ref. [1], App.A in [2])

1. Key definitions and examples: 
             -- parallel parking
             -- rolling ball on a rotating table
             -- golfer's dilemma

2. Classical mechanics, the Euler-Lagrange equations, Hamiltonian formalism.

3. Nonholonomic mechanics, the Lagrange-d'Alambert principle and equations.

4. Derivation of several nonholonomic systems from their Lagrangians

5. Symmetries, Noether theorem in mechanics. 
   Equations in vaconomic mechanics.



II. Subriemannian geometry (ref.[2])

6. The Carnot-Caratheodory metrics and their geodesics.
    Example of the Heisenberg group

7. The Chow-Rashevsky theorem: heuristics, proof.

8. Hausdorff measure, distribution rank, curvature, growth vector.

9. Normal and singular geodesics, examples, rigidity.

10. Interesting distributions: corank 1, Engel, Goursat, Carnot groups.

11. Applications: the falling cat, etc.


References:

 [1]. A. M. Bloch "Nonholonomic mechanics and control" 
      Interdisc. Appl. Math., vol.24. Springer-Verlag, New York, 2003, 483 pp.
 [2]. R. Montgomery "A tour of subriemannian geometries, their geodesics 
      and applications" Mathem. Surveys and Monographs, vol.91, 
      AMS, Providence, RI, 2002, 259 pp. 


Prerequisite(s): Some knowledge of Hamiltonian systems/symplectic geometry 
                 or MAT 1051HF (MAT 468H1F) is helpful.