Pfaffian equations: a variational perspective

Abstract

Pfaffian equations are part of a classic area well-established in Differential Geometry. From a more analytical viewpoint, it has been traditionally treated in textbooks dealing with Differential Equations focusing on analytical techniques to understand and, eventually, find solutions. The success of those methods are however limited to explicit situations where computations can be carried out to the end. There are some general existence and uniqueness results as well. Yet the numerical approximation of such solutions has not been, to the best of our knowledge, treated. By means of a variational approach based on a true vector variational problem, we propose a mechanism to examine and numerically approximate such solutions, explore its remarkable properties, and test its practical performance in a number of explicit examples. The method can be implemented in situations where the underlying field defining the Pfaffian equation is not known exactly, but may be part of a broader approximation scheme.