American Institute of Mathematical Sciences

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

Motivated by the work of Leznov-Mostovoy [17], we classify the linear deformations of standard \begin{document}$ 2n $\end{document}-dimensional phase space that preserve the obvious symplectic \begin{document}$ \mathfrak{o}(n) $\end{document}-symmetry. As a consequence, we describe standard phase space, as well as \begin{document}$ T^{*}S^{n} $\end{document} and \begin{document}$ T^{*}\mathbb{H}^{n} $\end{document} with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in \begin{document}$ {\mathbb{R}}^{n+2} $\end{document}.

The Piola identity \begin{document}$ \operatorname{div}\; \operatorname{cof} \;\nabla f = 0 $\end{document} is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of generalized variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.