François Gay-Balmaz

Dirac structures and reduction for nonholonomic mechanical systems on Lie groups with broken symmetry

We present the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The theory is illustrated with the help of finite and infinite dimensional examples. In particular, we show that the equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system.