A Geometric Explanation of Hamilton-Jacobi Methods Based on the Frobenius Theorem

XIAO Jing; LIU Chang; WANG Yong

doi:10.21656/1000-0887.370268

Volume 38 Issue 6

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XIAO Jing, LIU Chang, WANG Yong. A Geometric Explanation of Hamilton-Jacobi Methods Based on the Frobenius Theorem[J]. Applied Mathematics and Mechanics, 2017, 38(6): 708-714. doi: 10.21656/1000-0887.370268

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A Geometric Explanation of Hamilton-Jacobi Methods Based on the Frobenius Theorem

doi: 10.21656/1000-0887.370268

¹School of Information Technology, Guangdong Medical University, Dongguan, Guangdong 523808, P.R.China;²School of Physics, Liaoning University, Shenyang 110036, P.R.China;³State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology); Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

Funds: The National Natural Science Foundation of China（11572145;11202090）;China Postdoctoral Science Foundation（2014M560203）

Received Date: 2016-09-05
Rev Recd Date: 2016-09-30
Publish Date: 2017-06-15

Abstract

Abstract

With the differential geometry method, a geometric explanation based on the Frobenius theorem for characteristic equations of 1st-order partial differential equations was presented. According to the Frobenius theorem, the characteristic equations can be deduced directly from the 1st-order partial differential equations. Based on this, how to use the geometric method to find the corresponding Hamilton-Jacobi equations from Hamiltonian canonical equations was discussed. This method could be utilized to address the nonconservative or nonholonomic Hamiltonian mechanical problems. The classical Hamilton-Jacobi method is only a special case of this method.
- Hamilton-Jacobi theory,
- 1st-order partial differential equation,
- Frobenius theorem

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References(11)

References

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