Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed

The fundamental assumption of transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants (resp. products) must cross the surface only once and then proceed to products (resp. reactants). Recently it has been shown (Wiggins et al (2001) Phys. Rev. Lett. 86 5478; Uzer et al (2002) Nonlinearity 15 957) how to construct a dividing surface in phase space for Hamiltonian systems with an arbitrary (finite) number of degrees of freedom having the property that trajectories only cross once locally. In this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface.