American Institute of Mathematical Sciences

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincaré equations by several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle by several stages, extending methods known for the case of one stage in the previous literature. The case of Euler's disk is described as an illustrative example.

In this paper we explore the discretization of Euler-Poincaré-Suslov equations on SO(3), i.e. of the Suslov problem. We show that the consistency order corresponding to the unreduced and reduced setups, when the discrete reconstruction equation is given by a Cayley retraction map, are related to each other in a nontrivial way. We give precise conditions under which general and variational integrators generate a discrete flow preserving the constraint distribution. We establish general consistency bounds and illustrate the performance of several discretizations by some plots. Moreover, along the lines of [15] we show that any constraints-preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system.

We show that the Helmholtz conditions characterizing differential equations arising from variational problems can be expressed in terms of invariants of curves in a suitable Grassmann manifold.

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle we give a variational description of the equations of motion. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincaré and Lagrange Poincaré cases), variational problems for holomorphic maps, Sigma models or Chern-Simons theories. One of the advantages of our theory is that it is based in the existence of a multisymplectic form on a Lie algebroid.