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Exact Solution of the Problem of Brachistochrone with Allowance for the Coulomb Friction Forces

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Abstract

A system of differential equations describing the problem of brachistochrone with allowance for Coulomb friction forces is derived using the moving-basis method. An exact analytical solution of this system is obtained in the parametric form; a particular case of this solution in the limit \(\mu \to 0\) is transformed into the equation of classical brachistochrone.

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REFERENCES

  1. N. M. Günther, The Course of the Calculus of Variations (Gosthekhizdat, Moscow, 1941) [in Russian].

    Google Scholar 

  2. M. A. Lavrent’ev and L. A. Lyusternik, The Course of the Calculus of Variations, 2nd ed. (Gosthekhizdat, Moscow, 1950) [in Russian].

    Google Scholar 

  3. I. M. Gelfad and S. V. Fomin, Calculus of Variations (Fizmatgiz, Moscow, 1961; Dover, New York, 1963).

  4. V. I. Smirnov, Course of Higher Mathematics (Nauka, Moscow, 1967; Mir, Moscow, 1964), Vol. 4.

  5. L. E. Elsgoltz, Differential Equations and the Calculus of Variations (Nauka, Moscow, 1969; Mir, Moscow, 1977).

  6. N. Ashby, W. E. Brittin, W. F. Love, and W. Wyss, Am. J. Phys. 43, 902 (1975).

    Article  ADS  Google Scholar 

  7. S. C. Lipp, SIAM J. Control Optim. 35, 562 (1997).

    Article  MathSciNet  Google Scholar 

  8. J. C. Nayen, Int. J. Non-Lin. Mech. 40, 1057 (2005).

    Google Scholar 

  9. L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley, New York, 1962).

  10. S. O. Gladkov, Uch. Zap. Fiz. Fak. MGU, No. 4, 164002-1 (2016).

    Google Scholar 

  11. S. O. Gladkov and S. B. Bogdanova, J. Math. Sci. 245, 528 (2020).

    Article  Google Scholar 

  12. A. A. Barsuk and F. Paladi, Int. J. Non-Lin. Mech. 148, 43 (2023).

    Google Scholar 

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), 125993, Moscow, Russia

    S. O. Gladkov

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  1. S. O. Gladkov
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Correspondence to S. O. Gladkov.

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Translated by A. Sin’kov

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Gladkov, S.O. Exact Solution of the Problem of Brachistochrone with Allowance for the Coulomb Friction Forces. Tech. Phys. Lett. 49, 26–28 (2023). https://doi.org/10.1134/S1063785023040016

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  • DOI: https://doi.org/10.1134/S1063785023040016

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