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Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

Marsden, Jerrold E. and Wang, Hong and Zhang, Zhenxing (2014) Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry. Differential Geometry and its Applications, 33 . pp. 13-45. ISSN 0926-2245. https://resolver.caltech.edu/CaltechAUTHORS:20140612-085737235

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Abstract

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.difgeo.2014.01.003DOIArticle
http://www.sciencedirect.com/science/article/pii/S0926224514000047PublisherArticle
http://arxiv.org/abs/1202.3564arXivDiscussion Paper
Additional Information:© 2014 Elsevier B.V. Received 3 March 2012. Received in revised form 17 December 2012. Available online 3 February 2014. Communicated by T. S. Ratiu. The authors would like to thank the referees for their careful review reports and comments, which are helpful for us to revise well the manuscript. Especially grateful to Professor Tudor S. Ratiu and MS. Wendy McKay for their support and help. H. Wang's research was partially supported by the Natural Science Foundation of Tianjin (05YFJMJC01200) and the Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education, China.
Funders:
Funding AgencyGrant Number
Tianjin Natural Science Foundation05YFJMJC01200
Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education, ChinaUNSPECIFIED
Subject Keywords: Regular controlled Hamiltonian system; Symplectic structure; Momentum map; Regular Hamiltonian reduction; RCH-equivalence
Classification Code:MSC: 70H33; 53D20; 70Q05
Record Number:CaltechAUTHORS:20140612-085737235
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140612-085737235
Official Citation:Jerrold E. Marsden, Hong Wang, Zhenxing Zhang, Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry, Differential Geometry and its Applications, Volume 33, April 2014, Pages 13-45, ISSN 0926-2245, http://dx.doi.org/10.1016/j.difgeo.2014.01.003. (http://www.sciencedirect.com/science/article/pii/S0926224514000047)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:46226
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:12 Jun 2014 16:23
Last Modified:03 Oct 2019 06:42

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