Symmetry, stability, geometric phases and mechanical integrators

Marsden, J. E., O. M. O'Reilly, F. J. Wicklin and B. W. Zombro

Nonlinear Science Today, 1, 4-11, also 1, 14-21, (1991)

Abstract:

New analytical techniques and recent algorithms which numerically compute the time evolution of mechanical systems enable today's scientists, engineers, and mathematicians to predict events more accurately and more rapidly than ever before. Beyond the problems of simulation and prediction. However, lie the problems of understanding a dynamical system and choosing a correct dynamical system to model a given physical situation. Many systems remain too intricate to fully understand, but modern methods of mathematical analysis can sometimes offer insight. Most of this insight is obtained by viewing dynamics geometrically, and in fact the recent advances in mechanics which we review in this article all share this geometric perspective. Much of the value of these techniques lies in their applications, and although applications exist in a broad range of disciplines, we will focus on examples from space mechanics and robotics because these are simple to visualize.