American Institute of Mathematical Sciences

In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an \begin{document}$ n $\end{document}-dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an \begin{document}$ (n-1) $\end{document}-dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.

A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the Schrödinger form to classical mechanics.

In this paper, we show that the spaces of sections of the \begin{document}$ n $\end{document}-th differential operator bundle \begin{document}$ \mathfrak{D}^n E $\end{document} and the \begin{document}$ n $\end{document}-th skew-symmetric jet bundle \begin{document}$ \mathfrak{J}_n E $\end{document} of a vector bundle \begin{document}$ E $\end{document} are isomorphic to the spaces of linear \begin{document}$ n $\end{document}-vector fields and linear \begin{document}$ n $\end{document}-forms on \begin{document}$ E^* $\end{document} respectively. Consequently, the \begin{document}$ n $\end{document}-omni-Lie algebroid \begin{document}$ \mathfrak{D} E\oplus \mathfrak{J}_n E $\end{document} introduced by Bi-Vitagliano-Zhang can be explained as certain linearization, which we call pseudo-linearization of the higher analogue of Courant algebroids \begin{document}$ TE^*\oplus \wedge^nT^*E^* $\end{document}. On the other hand, we show that the omni \begin{document}$ n $\end{document}-Lie algebroid \begin{document}$ \mathfrak{D} E\oplus \wedge^n \mathfrak{J} E $\end{document} can also be explained as certain linearization, which we call Weinstein-linearization of the higher analogue of Courant algebroids \begin{document}$ TE^*\oplus \wedge^nT^*E^* $\end{document}. We also show that \begin{document}$ n $\end{document}-Lie algebroids, local \begin{document}$ n $\end{document}-Lie algebras and Nambu-Jacobi structures can be characterized as integrable subbundles of omni \begin{document}$ n $\end{document}-Lie algebroids.

In this work we introduce a category \begin{document}$ \mathfrak{L D P}_{d} $\end{document} of discrete-time dynamical systems, that we call discrete Lagrange–D'Alembert–Poincaré systems, and study some of its elementary properties. Examples of objects of \begin{document}$ \mathfrak{L D P}_{d} $\end{document} are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of \begin{document}$ \mathfrak{L D P}_{d} $\end{document} and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange–Poincaré systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in \begin{document}$ \mathfrak{L D P}_{d} $\end{document} to the system obtained by a one-stage reduction by the full symmetry group.

This paper extends the theory of controlled Hamiltonian systems with symmetries due to [23,9,10,6,7,11] to the case of non-abelian symmetry groups \begin{document}$ G $\end{document} and semi-direct product configuration spaces. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the \begin{document}$ G $\end{document}-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9,10]).