Journal of Geometric Mechanics最新文献

We consider a smooth begin{document}$ 2n $end{document}-manifold begin{document}$ M $end{document} endowed with a bi-Lagrangian structure begin{document}$ (omega,mathcal{F}_{1},mathcal{F}_{2}) $end{document}. That is, begin{document}$ omega $end{document} is a symplectic form and begin{document}$ (mathcal{F}_{1},mathcal{F}_{2}) $end{document} is a pair of transversal Lagrangian foliations on begin{document}$ (M, omega) $end{document}. Such a structure has an important geometric object called the Hess Connection. Among the many importance of Hess connections, they allow to classify affine bi-Lagrangian structures.

In this work, we show that a bi-Lagrangian structure on begin{document}$ M $end{document} can be lifted as a bi-Lagrangian structure on its trivial bundle begin{document}$ Mtimesmathbb{R}^n $end{document}. Moreover, the lifting of an affine bi-Lagrangian structure is also affine. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on begin{document}$ Mtimesmathbb{R}^n $end{document}. This lifting can be lifted again on begin{document}$ left(Mtimesmathbb{R}^{2n}right)timesmathbb{R}^{4n} $end{document}, and coincides with the initial dynamic (in our sense) on begin{document}$ Mtimesmathbb{R}^n $end{document}. By replacing begin{document}$ Mtimesmathbb{R}^{2n} $end{document} with the tangent bundle begin{document}$ TM $end{document} or cotangent bundle begin{document}$ T^{*}M $end{document} of begin{document}$ M $end{document}, results still hold when begin{document}$ M $end{document} is parallelizable.

We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie–Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie–Poisson systems. It also implies that solving the canonized system by symplectic Runge–Kutta methods yields so-called collective Lie–Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie–Poisson systems on begin{document}$ mathfrak{so}(2,1)^{*} $end{document} and begin{document}$ (mathfrak{se}(3) ltimes mathbb{R}^{3})^{*} $end{document}, respectively.

Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.

In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, bracket will be the generic term including product and brace. The path leads beyond binary to multi-linear begin{document}$ n $end{document}-ary operations, either for a single begin{document}$ n $end{document} or for whole coherent congeries of such assembled into what is known now as an begin{document}$ infty $end{document}-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.