American Institute of Mathematical Sciences

Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.

We consider the Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point with parameters in a complex domain. We suppose that these equations admit a first integral functionally independent of the three already known integrals which does not depend on all the variables. We prove that this may happen only in the already known three integrable cases or in the trivial case of kinetic symmetry. We provide a method for finding such a fourth integral, when it exists.

About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngọc by means of five invariants. Standard examples are the coupled spin oscillator on \begin{document}$\mathbb{S}^2 \times \mathbb{R}^2$\end{document} and coupled angular momenta on \begin{document}$\mathbb{S}^2 \times \mathbb{S}^2$\end{document}, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a \begin{document}$6$\end{document}-parameter family of integrable systems on \begin{document}$\mathbb{S}^2 \times \mathbb{S}^2$\end{document} and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.

The \begin{document}$\mathcal{KS}$\end{document} map is revisited in terms of an \begin{document}$S^1$\end{document}-action in \begin{document}$T^*\mathbb{H}_0$\end{document} with the bilinear function as the associated momentum map. Indeed, the \begin{document}$\mathcal{KS}$\end{document} transformation maps the \begin{document}$S^1$\end{document}-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous \begin{document}$S^1$\end{document}-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the \begin{document}$\mathcal{KS}$\end{document} transformations are needed.