Symmetric Reduction of Regular Controlled Lagrangian System with Momentum Map

2021/03/11 by Hong Wang, Wang, Hong · 1 citation
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #53D20 #70H33 #70Q05 #Control and Stability of Dynamical Systems #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Microtubule and mitosis dynamics #Symplectic Geometry (math.SG) #math.DG #math.DS #math.SG #msc:53D20 #msc:70H33 #msc:70Q05

paper · pdf · doi:10.48550/arxiv.2103.06563

26 pages. arXiv admin note: text overlap with arXiv:1802.01988, arXiv:1202.3564

arxiv created 2021/03/11 · openalex publication_date 2021/03/11 · arxiv updated 2021/03/12 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28

Abstract

In this paper, following the ideas in Marsden et al.[18], we set up the regular reduction theory of a regular controlled Lagrangian (RCL) system with symmetry and momentum map, by using Legendre transformation and Euler-Lagrange vector field, and this reduction is an extension of symmetric reduction theory of a regular Lagrangian system under regular controlled Lagrangian equivalence conditions. Considering the completeness of reduction, in order to describe uniformly the RCL systems defined on a tangent bundle and on its regular reduced spaces, we first define a kind of RCL systems on a symplectic fiber bundle. Then we give a good expression of the dynamical vector field of the RCL system, such that we can describe the RCL-equivalence for the RCL systems. Moreover, we introduce regular point and regular orbit reducible RCL systems with symmetries and momentum maps, by using the reduced Lagrange symplectic forms and the reduced Euler-Lagrange vector fields, and prove the regular point and regular orbit reduction theorems for the RCL systems and regular Lagrangian systems, which explain the relationships between RpCL-equivalence, RoCL-equivalence for the reducible RCL systems with symmetries and RCL-equivalence for the associated reduced RCL systems, as well as the relationship of equivalences of the regular reducible Lagrangian systems, R_p-reduced Lagrangian systems and R_o-reduced Lagrangian systems.

Citations

• The Geometrical Structure of Phase Space of the Controlled Hamiltonian System with Symmetry
• Regular Reduction of Controlled Hamiltonian System with Symplectic Structure and Symmetry

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