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Dynamics of a Pendulum in a Rarefied Flow

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Abstract

We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to the massive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle.

The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.

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REFERENCES

  1. Arnold, V. I., Ordinary Differential Equations, Berlin: Springer, 2006.

    Google Scholar 

  2. Newton, I., The Mathematical Principles of Natural Philosophy, Berkeley, Calif.: Univ. of California, 1999.

    Google Scholar 

  3. Buttazzo, G., Ferone, V., and Kawohl, B., Minimum Problems over Sets of Concave Functions and Related Questions, Math. Nachr., 1995, vol. 173, pp. 71–89.

    Article  MathSciNet  Google Scholar 

  4. Brock, F., Ferone, V., and Kawohl, B., A Symmetry Problem in the Calculus of Variations, Calc. Var. Partial Differential Equations, 1996, vol. 4, no. 6, pp. 593–599.

    Article  MathSciNet  Google Scholar 

  5. Lachand-Robert, Th. and Oudet, É., Minimizing within Convex Bodies Using a Convex Hull Method, SIAM J. Optim., 2005, vol. 16, no. 2, pp. 368–379.

    Article  MathSciNet  Google Scholar 

  6. Wachsmuth, G., The Numerical Solution of Newton’s Problem of Least Resistance, Math. Program. Ser. A, 2014, vol. 147, no. 1–2, pp. 331–350.

    Article  MathSciNet  Google Scholar 

  7. Plakhov, A., A Solution to Newton’s Least Resistance Problem Is Uniquely Defined by Its Singular Set, Calc. Var. Partial Differential Equations, 2022, vol. 61, no. 5, Paper No. 189, 37 pp.

    Article  MathSciNet  Google Scholar 

  8. Plakhov, A. and Tchemisova, T., Problems of Optimal Transportation on the Circle and Their Mechanical Applications, J. Differential Equations, 2017, vol. 262, no. 3, pp. 2449–2492.

    Article  MathSciNet  ADS  Google Scholar 

  9. Plakhov, A., Billiards and Two-Dimensional Problems of Optimal Resistance, Arch. Ration. Mech. Anal., 2009, vol. 194, no. 2, pp. 349–381.

    Article  MathSciNet  Google Scholar 

  10. Angel, O., Burdzy, K., and Sheffield, S., Deterministic Approximations of Random Reflectors, Trans. Amer. Math. Soc., 2013, vol. 365, no. 12, pp. 6367–6383.

    Article  MathSciNet  Google Scholar 

  11. Plakhov, A., Tchemisova, T., and Gouveia, P., Spinning Rough Disc Moving in a Rarefied Medium, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2010, vol. 466, no. 2119, pp. 2033–2055.

    MathSciNet  ADS  Google Scholar 

  12. Kryzhevich, S., Motion of a Rough Disc in Newtonian Aerodynamics, in EmC-ONS 2014: Optimization in the Natural Sciences, A. Plakhov, T. Tchemisova, A. Freitas (Eds.), Commun. Comput. Inf. Sci., vol. 499, Cham: Springer, 2015, pp. 3–19.

    Google Scholar 

  13. Rudzis, P., Rough Collisions, arXiv:2203.09102 (2022).

  14. Kryzhevich, S. and Plakhov, A., Billiard in a Rotating Half-Plane, J. Dyn. Control Syst., 2023, vol. 29, no. 4, pp. 1695–1707.

    Article  MathSciNet  Google Scholar 

  15. Kryzhevich, S. and Plakhov, A., Rotating Rod and Ball, J. Math. Anal. Appl., 2024, in press.

  16. Lokshin, B. Ya. and Samsonov, V. A., On a Heuristic Model of an Aerodynamical Pendulum, Fundam. Prikl. Mat., 1998, vol. 4, no. 3, pp. 1047–1061 (Russian).

    MathSciNet  Google Scholar 

  17. Shakhov, E. M., The Oscillations of a Satellite Probe Towed on an Inextensible Line in an Inhomogeneous Atmosphere, J. Appl. Math. Mech., 1988, vol. 52, no. 4, pp. 440–444; see also: Prikl. Mat. Mekh., 1988, vol. 52, no. 4, pp. 567-572.

    Article  Google Scholar 

  18. Zhuk, V. I. and Shakhov, E. M., Oscillations of a Tethered Satellite of Small Mass under the Effect of Aerodynamic and Gravitational Forces, Kosmicheskie Issledovaniya, 1990, vol. 28, no. 6, pp. 820–830 (Russian).

    ADS  Google Scholar 

  19. Lokshin, B. Ya., Samsonov, V. A., and Shamolin, M. V., Pendulum Systems with Dynamical Symmetry, J. Math. Sci. (N.Y.), 2017, vol. 227, no. 4, pp. 461–519.

    Article  MathSciNet  Google Scholar 

  20. Rybnikova, T. A. and Treshchev, D. V., Existence of Invariant Tori in the Problem of the Motion of a Satellite with a Solar Sail, Kosmicheskie Issledovaniya, 1990, vol. 28, no. 2, pp. 309–312 (Russian).

    ADS  Google Scholar 

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Funding

The work of AD was supported by the MSU Program of Development, Project No 23-SCH5-25. The work of AP was supported by the Center for R & D in Mathematics and Applications, refs. UIDB/04106/2020 and UIDP/04106/2020, and by CoSysM3, ref. 2022.03091.PTDC, through FCT.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Leninskie Gory 1, 119991, Moscow, Russia

    Alexey Davydov

  2. National University of Science and Technology MISIS, pr. Leninskiy, 19049, Moscow, Russia

    Alexey Davydov

  3. Center for R & D in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Alexander Plakhov

  4. Institute for Information Transmission Problems, per. Bolshoy Karetny 19, 127994, Moscow, Russia

    Alexander Plakhov

Authors
  1. Alexey Davydov
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  2. Alexander Plakhov
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Correspondence to Alexey Davydov or Alexander Plakhov.

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The authors declare that they have no conflicts of interest.

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MSC2010

34A34, 34C60, 70E17, 70G60

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Davydov, A., Plakhov, A. Dynamics of a Pendulum in a Rarefied Flow. Regul. Chaot. Dyn. 29, 134–142 (2024). https://doi.org/10.1134/S1560354724010088

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  • DOI: https://doi.org/10.1134/S1560354724010088

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