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Formalization of Euler–Lagrange Equation Set Based on Variational Calculus in HOL Light

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Abstract

As the theoretical foundation of Lagrangian mechanics, Euler–Lagrange equation sets are widely applied in building mathematical models of physical systems, especially in solving dynamics problems. However, their preconditions are often not fully satisfied in practice. Therefore, it is necessary to verify their applications. The purpose of the present work is to conduct such verification by establishing a formal theorem library of Lagrangian mechanics in HOL Light. For this purpose, some basic concepts such as functional variation and the necessary conditions for functional extreme are formalized. Then, the fundamental lemma of variational calculus is formally verified and some new constuctors and destructors are proposed. Finally, the Euler–Lagrange equation set is formalized. To validate the formalization, the formalization results are applied to verify the least resistance problem of gas flow. The present work not only lays a necessary and solid foundation for application involving Lagrangian mechanics but also extends the HOL Light theorem library.

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Acknowledgements

This work was supported by the the National Key R&D Plan (2017YFB1303000), National Natural Science Foundation of China (61876111, 61572331, 61602325, 61877040) and Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds (025185305000).

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Authors and Affiliations

  1. International Science and Technology Cooperation Base of Electronic System Reliability and Mathematical Interdisciplinary, Capital Normal University, Beijing, China

    Yong Guan & Jingzhi Zhang

  2. Beijing Engineering Research Center of High Reliable Embedded System, Capital Normal University, Beijing, China

    Guohui Wang

  3. Beijing Key Laboratory of Electronic System Reliability and Prognostics, College of Information Engineering, Capital Normal University, Beijing, China

    Ximeng Li & Zhiping Shi

  4. School of Mathematical Science, Capital Normal University, Beijing, China

    Jingzhi Zhang & Guohui Wang

  5. Beijing Advanced Innovation Center for Imaging Theory and Technology, Capital Normal University, Beijing, China

    Yongdong Li

  6. Capital Normal University, No. 105, West 3rd Ring North Road, Haidian District, Beijing, China

    Guohui Wang & Zhiping Shi

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  1. Yong Guan
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  2. Jingzhi Zhang
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  3. Guohui Wang
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  4. Ximeng Li
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Correspondence to Guohui Wang or Zhiping Shi.

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Guan, Y., Zhang, J., Wang, G. et al. Formalization of Euler–Lagrange Equation Set Based on Variational Calculus in HOL Light. J Autom Reasoning 65, 1–29 (2021). https://doi.org/10.1007/s10817-020-09549-w

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  • DOI: https://doi.org/10.1007/s10817-020-09549-w

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