On the third order helicity of magnetic fields on invariant domains in S 3

We introduce an alternative approach to the third order helicity of a volume preserving vector field X. The proposed approach exploits correspondence between the Milnor ¯ µ 123-invariant for 3-component links and the homotopy invariants of maps to configuration spaces, and we provide a simple geometric proof of this fact in the case of borromean links. Based on these connections we develop a formulation for the third order helicity of X on invariant unlinked domains of X, and provide Arnold's style er-godic interpretation of this invariant as an average asymptotic ¯ µ 123-invariant of orbits of X. We also obtain a lower bound for the L 2-energy of X.