American Institute of Mathematical Sciences

To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.

The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable \begin{document}$ (k_1,k_2,k_3) $\end{document}-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend on three parameters (\begin{document}$ k_1, k_2, k_3 $\end{document}) in such a way that in the particular case \begin{document}$ k_1\ne 0 $\end{document}, \begin{document}$ k_2 = k_3 = 0 $\end{document}, the properties characterizing the Kepler problem are obtained.

This paper can be considered as divided in two parts and every part presents a different approach (different complex functions and different quasi-bi-Hamil-tonian structures).

We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of mutually interacting (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the matched pair Lie-Poisson formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order \begin{document}$ \geq 2 $\end{document}. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.

The article presents a bundle framework for nonlinear observer design on a manifold with a a Lie group action. The group action on the manifold decomposes the manifold to a quotient structure and an orbit space, and the problem of observer design for the entire system gets decomposed to a design over the orbit (the group space) and a design over the quotient space. The emphasis throughout the article is on presenting an overarching geometric structure; the special case when the group action is free is given special emphasis. Gradient based observer design on a Lie group is given explicit attention. The concepts developed are illustrated by applying them on well known examples, which include the action of \begin{document}$ {\mathop{\mathbb{SO}(3)}} $\end{document} on \begin{document}$ \mathbb{R}^3 \setminus \{0\} $\end{document} and the simultaneous localisation and mapping (SLAM) problem.

Erratum note for "Constraint algorithm for singular field theories in the \begin{document}$ k $\end{document}-cosymplectic framework".