Quasi-canonicalization for linear homogeneous nonholonomic systems

doi:10.1088/1674-1056/ab8627

中国物理B ›› 2020, Vol. 29 ›› Issue (6): 64501-064501.doi: 10.1088/1674-1056/ab8627

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

Quasi-canonicalization for linear homogeneous nonholonomic systems

Yong Wang(王勇), Jin-Chao Cui(崔金超), Ju Chen(陈菊), Yong-Xin Guo(郭永新)

1 School of Biomedical Engineering, Guangdong Medical University, Dongguan 523808, China;
2 School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;
3 College of Physics, Liaoning University, Shenyang 110036, China

收稿日期:2020-02-08 修回日期:2020-03-13 出版日期:2020-06-05 发布日期:2020-06-05
通讯作者: Yong-Xin Guo E-mail:yxguo@lnu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11972177, 11972122, 11802103, 11772144, 11872030, and 11572034) and the Scientific Research Starting Foundation for Scholars with Doctoral Degree of Guangdong Medical University (Grant Nos. B2019042 and B2019021).

Quasi-canonicalization for linear homogeneous nonholonomic systems

Yong Wang(王勇)¹, Jin-Chao Cui(崔金超)¹, Ju Chen(陈菊)², Yong-Xin Guo(郭永新)³

1 School of Biomedical Engineering, Guangdong Medical University, Dongguan 523808, China;
2 School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;
3 College of Physics, Liaoning University, Shenyang 110036, China

Received:2020-02-08 Revised:2020-03-13 Online:2020-06-05 Published:2020-06-05
Contact: Yong-Xin Guo E-mail:yxguo@lnu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11972177, 11972122, 11802103, 11772144, 11872030, and 11572034) and the Scientific Research Starting Foundation for Scholars with Doctoral Degree of Guangdong Medical University (Grant Nos. B2019042 and B2019021).

摘要/Abstract

摘要： For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπ^μ∧dξ_μ, in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates π^μ and quasi-momenta ξ_μ. The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates π^μ by a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton-Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton-Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton-Jacobi method.

关键词: quasi-canonicalization, nonholonomic systems, first-order linear mapping, Hamilton-Jacobi method

Abstract: For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπ^μ∧dξ_μ, in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates π^μ and quasi-momenta ξ_μ. The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates π^μ by a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton-Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton-Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton-Jacobi method.

Key words: quasi-canonicalization, nonholonomic systems, first-order linear mapping, Hamilton-Jacobi method

中图分类号: (Lagrangian and Hamiltonian mechanics)

45.20.Jj

02.40.Yy (Geometric mechanics) 45.50.-j (Dynamics and kinematics of a particle and a system of particles) 02.40.Ky (Riemannian geometries)

王勇, 崔金超, 陈菊, 郭永新. Quasi-canonicalization for linear homogeneous nonholonomic systems[J]. 中国物理B, 2020, 29(6): 64501-064501.

Yong Wang(王勇), Jin-Chao Cui(崔金超), Ju Chen(陈菊), Yong-Xin Guo(郭永新). Quasi-canonicalization for linear homogeneous nonholonomic systems[J]. Chin. Phys. B, 2020, 29(6): 64501-064501.

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Quasi-canonicalization for linear homogeneous nonholonomic systems

Quasi-canonicalization for linear homogeneous nonholonomic systems

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