Geometric integration: numerical solution of differential equations on manifolds

Abstract

Since their introduction by Sir Isaac Newton, diffierential equations have played a decisive role in the mathematical study of natural phenomena. An important and widely acknowledged lesson of the last three centuries is that critical information about the qualitative nature of solutions of diffierential equations can be determined by studying their geometry. Perhaps the most important example of this approach was the formulation of the laws of mechanics by Alexander Rowan Hamilton, which allowed deep geometric tools to be used in understanding the dynamics of complex systems such as rigid bodies and the Solar System. Conserved quantities of a Hamiltonian system, such as energy, linear and angular momentum, could be understood in terms of the symmetries of the underlying Hamiltonian function, its ergodic properties determined from the underlying symplectic nature of the formulation and constraints on the system could be incorporated in a natural manner. The Hamiltonian geometric formulation of many other problems in science modelled by ordinary and partial diffierential equations, such as ocean dynamics, nonlinear optics and elastic deformations, continues to play a vital role in our qualitative understanding of these systems. An equally important geometric approach to the study of diffierential equations is the application of symmetry–based methods pioneered by Sophus Lie. Exploiting underlying symmetries of a partial or ordinary difierential equation, it can be often greatly simplified and sometimes solved altogether in closed form. Such methods, which lie at the heart of the construction of self–similar solutions of diffierential equations and the symmetry reduction of complex systems, have become increasingly popular with the development of symbolic algebra packages. It is no coincidence that the most important equations of mathematical physics are precisely those for which geometric and symmetry–based methods are most effiective. Arguably, these equations are really a shorthand for the deep underlying symmetries in nature that they encapsulate.