First integrals of Generalized Darboux-Halphen systems and Membrane Paradigm

Abstract

The Darboux-Halphen system of equations have common or individual additive terms depending on the matrices defining Yang-Mills gauge potential fields. Tod (Phys. Lett. A 190 (1994) 221-224), described a conserved quantity for the classical systems with no additive terms. We show that the conserved quantity apply even for the generalized cases with common additive terms. A theory has been presented, with an example, of how to formulate conserved quantities for equation with individual additive terms. We also briefly shed some light on the issues of surface motions of fluids in conncetion to Nahm`s equation and the self-duality and integrability of membrane dynamics.