Life without (variational) principles: non-holonomic systems

Prof. Jair Koiller , National Laboratory for Scientific Computation in Brazil, Visiting Professor of CDS

In the first part of the talk, we discuss some of the differences between vak-onomic (sub-riemannian) and non-holonomic systems. After that, we present the main results of a short communication by E. Cartan at the 1928 IMU Congress. Cartan proposed using his moving frame method to classify nonholonomic systems $(L = T - V, E)$ on $TQ$, where $E \subset TQ$ is the constraint distribution. He essentially solved the problem for $[E,E] = TQ$. We discuss (but not solve) the case of higher nonholonomicity degree. Finally, in another tack, we discuss the best scenario for reduction by symmetries. Namely, when $(G \hookrightarrow Q \rightarrow S)$ is a principal bundle and $E$ is the horizontal distribution of a connection. Then $T^*S$ inherits a "compressed" almost Poisson bracket dynamics, which is non-degenerate. In some cases it can (but not always) be conformally Poisson. In some cases, there is a smooth invariant measure. The last part is joint work with P.P. Rios and K. Ehlers.