Title:
"Geometric Mechanics and sub-Riemannian geometry"
Lecturer:
Richard Montgomery (UCSC)
1. Metrics vs Cometrics.
A) Mechanics on the line. Hamiltionian and Lagrangian.
B) Mechanics on Euclidean spaces: Hamiltonian and Lagrangian;
C) Mechanics on manifolds. Hamiltonian and Lagrangian
D) The three ways to write Riemannian geodesics: via Levi-Civita connection,
via the Lagrangian, or via the Hamiltonian
E) the one available method for subRiemannian: Hamiltonian
[15 min to 1/2 hour per topic ]
I aim to use the Heisenberg group , and its spherical incarnation to illustrate.
With luck, symplectic forms, the Legendre transformation and Poisson brackets are discussed.
2. Falling cats & principal bundles.
A cat, falling from upside down, is faced with a mechanics problem which at the same time is the basic problem in sub-Riemannian geometry: how to change her shape in such a way to land on her feet.
There is a velocity constraint at each instant: ``angular momentum equals zero''.
Angular momentum is associated to the action of the rotation group on the cat's configuration space.
In the language of Riemannian geometry this action is one by isometries and the angular momentum zero constraint asserts that the velocity (to configuration space) is orthogonal to the group orbit. The angular momentum zero constraint defines a non-integrable distribution (linear sub-bundle of the tangent bundle) and kinetic energy defines a metric so that the falling cat's problem becomes the problem of finding a subRiemannian geodesic between the two points ``upside down'' and ``right-side up''.
In the first half I start to set up the cat's problem as a problem in subRiemannian geometry.
As a problem within sR geometry it fits within the general context of ``isoholonomic problems'':
among all loops on the base space of a principal bundle having a given holonomy find the shortest.
In part 2 of this lecture I go through some of the basics of principal bundle geometry, and of Riemannian submersions, viewing principal bundles as free G-spaces.
3. Abnormal extremals.
We reformulate the cat's problem as the problem of a (possibly non-Abelian) charged particle in a (non-Abelian') magnetic field and discuss some intuition around abnormal extremals. I focus on the Abelian case as giving the simplest instances of singular minimizers.
4. Poisson and Symplectic Reduction.
We describe some general techniques of reducing by symmetries which allow one to explicitly compute sR geodesics in certain contexts. This comes from ch 13 of my book.
5. Two open problems.
What remains to do; Lie theoretic computations.
Are all minimizing sR geodesics smooth? Does Sard hold for the endpoint map?
We will discuss where these problems came from and what seems to be the known and where the obstacles lie.