A THEOREM OF BÄCKLUND REVISITED

Abstract

Bäcklund's theorem states that the most general contact transformation is an extended point transformation whenever both the number of independent variables and the number of dependent variables exceed one. A partial circumvention of Bäcklund's theorem is obtained by assigning each dependent variable its own distinct manifold of independent variables. This gives rise to extended symplectic product structures. sequences of extended Hamiltonians, and Lie groups of regular maps that satisfy systems of extended Hamilton-Jacobi equations provided the initial data is determined by a regular map. These ideas are applied to the study of systems of nonlinear second order partial differential equations. Lie groups of solutions are shown to be obtained by solving systems of extended Hamilton-Jacobi equations provided the initial data defines a solution.