Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 12, 2018
11:10 a.m.

PhD Candidate:  Boris Khesin
Supervisor:   Daniel Fusca
Thesis title:  A groupoid approach to geometric mechanics



In 1966 V. Arnold proved that the Euler equation for an incompressible fluid describes the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms of the fluid’s domain. This remarkable observation led to numerous advances in the study of the Hamiltonian properties, instabilities, and topological features of fluid flows. However, Arnold’s approach does not apply to systems whose configuration spaces do not have a group structure. A particular example of such a system is that of a fluid with moving boundary. More generally, one can consider a system describing a rigid body moving in a fluid. Here the configurations of the fluid are identified with diffeomorphisms mapping a fixed reference domain to the exterior of the (moving) body. In general such diffeomorphisms cannot be composed, since the domain of one will not match the range of the other.

The systems we consider are numerous variations of a rigid body in an inviscid fluid. The different cases are specified by the properties of the fluid; the fluid may be compressible or incompressible, irrotational or not. By using groupoids we generalize Arnold’s diffeomorphism group framework for fluid flows to show that the well-known equations governing the motion of these various systems can be viewed as geodesic equations (or more generally, Newton’s equations) written on an appropriate configuration space.

We also show how constrained dynamical systems on larger algebroids are in many cases equivalent to dynamical systems on smaller algebroids, with the two systems being related by a generalized notion of Riemannian submersion. As an application, we show that incompressible fluid-body motion with the constraint that the fluid velocity is curl- and circulation-free is equivalent to solutions of Kirchhoff’s equations on the finite-dimensional algebroid $\mathfrak{se}(n)$.

In order to prove these results, we further develop the theory of Lagrangian mechanics on algebroids. Our approach is based on the use of vector bundle connections, which leads to new expressions for the canonical equations and structures on Lie algebroids and their duals.

The case of a compressible fluid is of particular interest by itself. It turns out that for a large class of potential functions $U$, the gradient solutions of the compressible fluid equations can be related to solutions of Schr\”{o}dinger-type equations via the $\emph{Madelung transform}$, which was first introduced in 1927. We prove that the Madelung transform not only maps one class of equations to the other, but it also preserves the Hamiltonian properties of both equations.

A copy of the thesis can be found here:  fusca thesis


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