| Contents |
| We consider a general approach for the process of Lagrangian and Hamiltonian
reduction by symmetries in condensed matter. This approach is used to show
the complete integrability of several one dimensional texture equations arising in
liquid Helium phases and neutron stars.
The key to the success of our geometric method is the fact that all
physical systems under study have a natural Lagrangian and
Hamiltonian formulation within the Lagrange-Poincar\'e and
Hamilton-Poincar\'{e} theories, with the Lagrangian and Hamiltonian
independent on a very special group of variables. This implies
that these systems have an equivalent Euler-Poincar\'e and
Lie-Poisson description which turns out to be considerably
simpler and more appropriate to the study of the dynamics of the
equations associated to the relevant phases. The possibility of
using at once the four descriptions of the systems under consideration
leads directly to the proof of complete integrability of the
equations describing the system's behavior in different phases. |
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