  The connection between swimming and control theory is attracting increasing
attention in the recent literature. Starting from an idea of Alberto Bressan
[7] we study the system of a planar body whose position and shape are described
by a finite number of parameters, and is immersed in a 2-dimensional ideal and
in-compressible fluid in terms of gauge field on the space of shapes. We focus
on a class of deformations measure preserving which are diffeomeorphisms whose
existence is ensured by the Riemann Mapping Theorem. We face a crucial problem:
the pres-ence of possible non vanishing initial impulse. If the body starts
with zero initial impulse we recover the results present in literature
(Marsden, Munnier and oths). If instead the body starts with an initial impulse
different from zero, the swimmer can self-propel in almost any direction if it
can undergo shape changes without any bound on their velocity. This interesting
observation, together with the analysis of the controllability of this system,
seems innovative.
