Weak-Hamiltonian dynamical systems
Weak-Hamiltonian dynamical systems
复制标题
弱哈密尔顿动力系统
DOI：
10.1063/1.2769145
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发表时间：
2007
影响因子：
1.3
通讯作者：
I. Vaisman
中科院分区：
文献类型：
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作者：
I. Vaisman
一个大的迷向结构E是TM * T*M的迷向子丛，它被赋予由配对定义的度量。结构E称为可积的，如果柯朗括号[X，Y]<$ΓE，<$X，Y <$ΓE。然后，必然地，人们还具有[X，Z]<$ΓE <$，<$Z <$ΓE <$[Vaisman，I.，“Isotropic subbundles of TM_T*M，”Int. J. Geom. Methods Mod.Phys.4，487-516（2007）]。弱哈密顿动力系统是一个向量场XH，使得（XH，dH）<$ΓE <$（H <$C∞（M））。在正则性条件dim（prT*ME）= const下，我们得到了XH的显式表达式和E的可积性条件，证明了端口控制的Hamilton系统（特别是约束力学）[Dalsmo，M.和货车der Schaft，A. J.，“关于能量守恒物理系统中数学结构的表示和可积性”，SIAM J. Control Optim。37，54-91（1998）]可以解释为弱哈密顿系统。最后，我们给出了弱Hamilton系统的约化定理和约束力学系统的一个相应推论。
A big-isotropic structure E is an isotropic subbundle of TM⊕T*M, endowed with the metric defined by pairing. The structure E is said to be integrable if the Courant bracket [X,Y]∊ΓE, ∀X,Y∊ΓE. Then, necessarily, one also has [X,Z]∊ΓE⊥, ∀Z∊ΓE⊥ [Vaisman, I., “Isotropic subbundles of TM⊕T*M,” Int. J. Geom. Methods Mod. Phys. 4, 487–516 (2007)]. A weak-Hamiltonian dynamical system is a vector field XH such that (XH,dH)∊ΓE⊥(H∊C∞(M)). We obtain the explicit expression of XH and of the integrability conditions of E under the regularity condition dim(prT*ME)=const. We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) [Dalsmo, M. and van der Schaft, A. J., “On representations and integrability of mathematical structures in energy conserving physical systems,” SIAM J. Control Optim. 37, 54–91 (1998)] may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems.