ON THE INTEGRABILITY AND PERTURBATION OF 3-DIMENSIONAL FLUID-FLOWS WITH SYMMETRY

The purpose of this paper is to develop analytical methods for studyin g particle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensional volume preserving vect or fields that are invariant under the action of a one-parameter symme try group whose infinitesimal generator is autonomous and volume-prese rving. We show that there exists a coordinate system in which the vect or field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedo m Hamiltonian system with the evolution of the remaining coordinate be ing governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they are field-i ndependent. The coordinate transformation is constructive. If the vect or field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced to action-an gle-angle coordinates. These are analogous to the familiar action-angl e variables from Hamiltonian mechanics and are quite useful for pertur bative studies of the class of sytems we consider. In fact, we show ho w our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dime nsional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flo ws considered, and draw an analogy with Clebsch variables in fluid mec hanics.