Applied Maths Seminar: James Meiss -- Generating Volume Preserving Maps

SPEAKER: James Meiss, Department of Applied Mathematics, University of Colorado at
Boulder 

LOCATION: *Carslaw 173* **please notice the unusual room** 

Abstract: It is common in Hamiltonian mechanics to use implicit generating functions to
obtain canonical transformations or equivalently symplectic maps.  These functions are
used in perturbation theory, in the construction of symplectic integrators, in the
variational formalism of Aubry and Mather, and in the computation of symplectic fluxes.
In this talk I will show how a similar formalism applies to the volume preserving case.
Such maps are of interest, for example, in fluid mechanics since the flow of an
incompressible fluid is volume preserving, as well as to plasma physics, since the
divergence of the magnetic field vanishes.  

The generators for canonical transformations are functions, but here we think of these
as zero-forms.  In the volume preserving maps the generators are also differential
forms, but now higher dimensional; for example, one-forms generate a three-dimensional
map.  Just as in the canonical case, our generators are implicit, and the resulting maps
must satisfy some "twist" conditions.  

We have used our generators to obtain formulas for lobe volumes that are needed to
compute transport.  We propose that accurate volume preserving integrators may be
constructed and that perturbation theory may be formulated using these forms.  But these
are problems for the future.