An index for closed orbits in Beltrami fields
Abstract
We consider the class of Beltrami fields (eigenfields of the curl operator) on threedimensional Riemannian solid tori: such vector fields arise as steady incompressible inviscid fluids and plasmas. Using techniques from contact geometry, we construct an integervalued index for detecting closed orbits in the flow which are topologically inessential (they have winding number zero with respect to the solid torus). The most important feature of the index is its computability: it can be rigorously determined from a C^{1}approximation to the vector field on any meridional disc of the solid torus. As a result, we obtain a test for detecting the nonexistence of a global crosssection to the 3D flow via purely 2D information.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 November 2001
 DOI:
 10.1016/S01672789(01)003438
 arXiv:
 arXiv:math/0101095
 Bibcode:
 2001PhyD..159..180E
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematical Physics;
 76;
 37N10
 EPrint:
 doi:10.1016/S01672789(01)003438