Canonical structure and symmetries of the Schlesinger equations

Lie symmetries of the canonical connection: codimension one abelian nilradical case

This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The conditions that determine a Lie symmetry in such a case are completely integrated. Finally the results obtained are compared with some four-dimensional Lie groups whose Lie algebras have three-dimensional abelian ni…

Symmetries and conservation laws of Hamiltonian systems

In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid vector fields and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, an example from optimal control theory is given.

内部拉格朗日量与空间规范对称性

Internal Lagrangians and spatial-gauge symmetries

本文利用无限延拓微分方程的内蕴几何重构哈密顿形式体系，引入空间方程与空间规范对称性概念，提出并演示非协变及协变正则变分原理，适用于任意变分方程。

Symmetries of the space of connections on a principal G-bundle and related symplectic structures

We investigate G-invariant symplectic structures on the cotangent bundle T*P of a principal G-bundle P(M,G) which are canonically related to automorphisms of the tangent bundle TP covering the identity map of P and commuting with the action of TG on TP. The symplectic structures corresponding to connections on P(M,G) are also investigated. The Marsden-Weinstein reduction procedure for these symplectic structures is discussed.

CMC hierarchy I: Commuting symmetries and loop algebra

We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loop algebra valued Gauß$\,$ map, the CMC hierarchy is obtained by the assembly of a pair of Adler-Kostant-Symes bi-Hamiltonian hierarchies to the original CMC system. The infinite sequence of higher-order conservation laws of the CMC system admits the corresponding extension, and w…

Hamiltonian Algebroid Symmetries in W-gravity and Poisson sigma-model

Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize the Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra c…