A solution method for decomposing vector fields in Hamilton energy

doi:10.1088/1674-1056/ad5a74

中国物理B ›› 2024, Vol. 33 ›› Issue (9): 98702-098702.doi: 10.1088/1674-1056/ad5a74

A solution method for decomposing vector fields in Hamilton energy

Xin Zhao(赵昕), Ming Yi(易鸣), Zhou-Chao Wei(魏周超), Yuan Zhu(朱媛), and Lu-Lu Lu(鹿露露)†

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

收稿日期:2024-05-13 修回日期:2024-06-14 接受日期:2024-06-21 发布日期:2024-08-15
通讯作者: Lu-Lu Lu E-mail:lululu@cug.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12305054, 12172340, and 12371506)

A solution method for decomposing vector fields in Hamilton energy

Xin Zhao(赵昕), Ming Yi(易鸣), Zhou-Chao Wei(魏周超), Yuan Zhu(朱媛), and Lu-Lu Lu(鹿露露)†

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Received:2024-05-13 Revised:2024-06-14 Accepted:2024-06-21 Published:2024-08-15
Contact: Lu-Lu Lu E-mail:lululu@cug.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12305054, 12172340, and 12371506)

摘要/Abstract

摘要： Hamilton energy, which reflects the energy variation of systems, is one of the crucial instruments used to analyze the characteristics of dynamical systems. Here we propose a method to deduce Hamilton energy based on the existing systems. This derivation process consists of three steps: step 1, decomposing the vector field; step 2, solving the Hamilton energy function; and step 3, verifying uniqueness. In order to easily choose an appropriate decomposition method, we propose a classification criterion based on the form of system state variables, i.e., type-I vector fields that can be directly decomposed and type-I$\!$I vector fields decomposed via exterior differentiation. Moreover, exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition. Finally, we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior. This solution provides a new approach for deducing the Hamilton energy function, especially in high-dimensional systems.

关键词: Hamilton energy, dynamical systems, vector field, exterior differentiation

Abstract: Hamilton energy, which reflects the energy variation of systems, is one of the crucial instruments used to analyze the characteristics of dynamical systems. Here we propose a method to deduce Hamilton energy based on the existing systems. This derivation process consists of three steps: step 1, decomposing the vector field; step 2, solving the Hamilton energy function; and step 3, verifying uniqueness. In order to easily choose an appropriate decomposition method, we propose a classification criterion based on the form of system state variables, i.e., type-I vector fields that can be directly decomposed and type-I$\!$I vector fields decomposed via exterior differentiation. Moreover, exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition. Finally, we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior. This solution provides a new approach for deducing the Hamilton energy function, especially in high-dimensional systems.

Key words: Hamilton energy, dynamical systems, vector field, exterior differentiation

中图分类号: (Neuroscience)

87.19.L-

87.19.lj (Neuronal network dynamics) 05.45.-a (Nonlinear dynamics and chaos) 05.45.Pq (Numerical simulations of chaotic systems)

Xin Zhao(赵昕), Ming Yi(易鸣), Zhou-Chao Wei(魏周超), Yuan Zhu(朱媛), and Lu-Lu Lu(鹿露露). A solution method for decomposing vector fields in Hamilton energy[J]. 中国物理B, 2024, 33(9): 98702-098702.

Xin Zhao(赵昕), Ming Yi(易鸣), Zhou-Chao Wei(魏周超), Yuan Zhu(朱媛), and Lu-Lu Lu(鹿露露). A solution method for decomposing vector fields in Hamilton energy[J]. Chin. Phys. B, 2024, 33(9): 98702-098702.

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A solution method for decomposing vector fields in Hamilton energy

A solution method for decomposing vector fields in Hamilton energy

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