Isometric Immersions and the Waving of Flags

Abstract

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values into R 3 satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. The submanifold result is then used to derive the equations of motion, after equipping the space of isometric immersions with its natural kinetic energy. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.