This requires a study of the structure of Poisson algebras of classical systems and their representations. Communicated by Stephen Wiggins Summary. We investigate here the consequences of relaxing the restrictions of the form of the coordinate transformations. When the linearized problem is autonomous with diagonal Jordan form, we can remove all nonlinearities order by order using these singular coordinate transformations. Communicated by Jerrold Marsden and Stephen Wiggins Summary.
Stokesian swimming is a geometric exercise, a collective game.
In Part I, we review Shapere and Wilczek's gauge-theoretical approach for a single organism. We estimate the speeds of organisms moving by propagating small amplitude waves, and we make a conjecture regarding a new inequality for the Stokes' curvature. In Part II, we extend the gauge theory to collective motions. We advocate the influx of nonlinear control theory and subriemannian geometry. Computationally, parallel algorithms are natural, each microorganism representing a separate processor.