Physics – Mathematical Physics
Scientific paper
2011-09-14
Physics
Mathematical Physics
28 pages; 5 figures
Scientific paper
In this paper, we consider the motion of the hydrodynamic Chaplygin sleigh, a planar rigid body in a potential flow with circulation around the body, subject to a nonholonomic constraint modeling a fin or keel attached to the body. We show that the motion of this system can be described by Euler-Poincare-Suslov equations on a central extension of the special Euclidian group SE(2), where the cocycle used to construct the extension encodes the effects of circulation upon the body. In the second part of the paper, we then discuss nonholonomic systems on central extensions of Lie groups, where both the Lagrangian and the nonholonomic constraints are left invariant. We show that there is a one-to-one correspondence between invariant measures on the original group and on the extended group, and we use this result to characterize the existence of an invariant measure for the hydrodynamic Chaplygin sleigh. We finish with a qualitative discussion of the reduced dynamics.
Garcia-Naranjo Luis C.
Vankerschaver Joris
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