Производящие функции оператора Коши гамильтоновой системы

Alexander S. Shmyrov; Vasiliy A. Shmyrov; Dmitry V. Shymanchuk

doi:10.21638/11701/spbu10.2023.408

Authors

Alexander S. Shmyrov St. Petersburg State University, 199034, St. Petersburg, Russian Federation
Vasiliy A. Shmyrov St. Petersburg State University, 199034, St. Petersburg, Russian Federation
Dmitry V. Shymanchuk St. Petersburg State University, 199034, St. Petersburg, Russian Federation

DOI:

https://doi.org/10.21638/11701/spbu10.2023.408

Abstract

The article is related to the mathematical apparatus for describing the phase trajectories of a hamiltonian system. An approach related to the construction of generating functions for the Cauchy operator is proposed. It is shown that one-parameter families of generating functions satisfy the Hamilton — Jacobi equation or its modifications. Using the example of small oscillations of a mathematical pendulum, it is shown that the description of the Cauchy operator for sufficiently long periods of time requires the use of generating functions of various types. With the help of generating functions, a variational principle similar to the principle of least action is formulated. The efficiency of using generating functions in the development of conservative methods of numerical integration is also noted.

Keywords:

Hamilton equations, generating function, Cauchy operator, variational principle

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References

Литература

Арнольд В. И. Математические методы классической механики. М.: Наука, 1989. 472 с.

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Shmyrov A., Shmyrov V., Shymanchuk D. The research of motion in a neighborhood of collinear libration point by conservative methods // AIP Conference Proceedings. 9^th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences. AMiTaNS 2017. 2017. Vol. 1895. Art. N 060003. htpps://doi.org/10.1063/1.5007388

Малявкин Г. П., Шмыров А. С., Шмыров В. А. Об одном численном методе для управляемых гамильтоновых систем // Вестник Санкт-Петербургского государственного университета технологии и дизайна. Естественные и технические науки. 2016. № 2. С. 34–37.

References

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Yoshida H. Construction of higher order symplectic integrators. Physics Letters A, 1990, vol. 150, iss. 5–7, pp. 262–268. https://doi.org/10.1016/0375-9601(90)90092-3

Shmyrov A., Shmyrov V., Shymanchuk D. The research of motion in a neighborhood of collinear libration point by conservative methods. AIP Conference Proceedings. 9^th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences, AMiTaNS 2017, 2017, vol. 1895, art. no. 060003. https://doi.org/10.1063/1.5007388

Malyavkin G. P., Shmyrov A. S., Shmyrov V. A. Ob odnom chislennom metode dlya upravlyayemykh gamil'tonovykh sistem [On the numerical method for a controlled hamiltonian system]. Vestnik of Saint Petersburg State University of Technology and Design. Natural and Technical Science, 2016, no. 2, pp. 34–37. (In Russian)