Reduction and reconstruction aspects of second-order dynamical systems with symmetry

Symmetry reduction, integrability and reconstruction in k-symplectic field theory

We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a k-connection to provide a reconstruction method. We show …

Nonlocal aspects of $λ$-symmetries and ODEs reduction

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $λ$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $λ$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $λ$, any $λ$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero f…

Reduction and reconstruction of multisymplectic Lie systems

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie s…

Not-quite-Hamiltonian reduction

The not-quite-Hamiltonian theory of singular reduction and reconstruction is described. This includes the notions of both regular and collective Hamiltonian reduction and reconstruction.

源自 Goursat 分布的新 Lie 系统：约化与重构

New Lie systems from Goursat distributions: reductions and reconstructions

本文证明了几类括号生成分布能导出具有相容几何结构的新 Lie 系统类，以 n-trailer 系统为例展示其对称约化与解的重构机制，为研究更广泛的微分方程提供了新思路。

The GIT aspect of generalized K$\ddot{a}$hler reduction. I

We revisit generalized K$\ddot{a}$hler reduction introduced by Lin and Tolman in \cite{LT} from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary K$\ddot{a}$hler reduction can be generalized without much effort to the generalized setting. It is also shown how generalized holomorphic structures arise naturally from the reduction procedure.