Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread

    2019, Vol. 15, no. 4, pp.  513-523

    Author(s): Kozlov V. V.

    It is well known that the maximal value of the central moment of inertia of a closed homogeneous thread of fixed length is achieved on a curve in the form of a circle. This isoperimetric property plays a key role in investigating the stability of stationary motions of a flexible thread. A discrete variant of the isoperimetric inequality, when the mass of the thread is concentrated in a finite number of material particles, is established. An analog of the isoperimetric inequality for an inhomogeneous thread is proved.
    Keywords: moment of inertia, Sundman and Wirtinger inequalities, articulated polygon
    Citation: Kozlov V. V., Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  513-523

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    [1] Appell, P., Traité de mécanique rationelle: Vol. 1. Statique. Dynamique du point, 6th ed., Gauthier-Villars, Paris, 1941, 612 pp.  mathscinet
    [2] Vesnitskii, A. I., Waves in Systems with Moving Boundaries and Loads, Nauka, Fizmatlit, Moscow, 2001, 320 pp. (Russian)
    [3] Yakubovskii, Yu. V., Zhivov, V. S., Koritysskii, Ya. I., and Migushov, I. I., Fundamentals of Yarn Mechanics, Lyogkaya Industriya, Moscow, 1973, 271 pp. (Russian)
    [4] Biggins, J. S. and Warner, M., “Understanding the Chain Fountain”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470:2163 (2014), 20130689  crossref  adsnasa
    [5] Gyulamirova, N. S. and Kugushev, E. I., “Stationary Form of a Moving Heavy Flexible Thread”, Mosc. Univ. Mech. Bull., 73:1 (2018), 7–10  mathnet  crossref  zmath; Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 2018, no. 1, 39–43 (Russian)  zmath
    [6] Pólya, G. and Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stud., 27, Princeton Univ. Press, Princeton, N.J., 1951, xvi+279 pp.  mathscinet
    [7] Kuz'min, P. A., “On the Stability of a Circular Shape of a Flexible Thread”, Trudy KAI, 20 (1948), 69–91 (Russian)
    [8] Kuz'min, P. A., “Stability of a Circular Shape of a Flexible Thread Having a Countable Set of Degrees of Freedom”, Trudy KAI, 22 (1949), 3–15 (Russian)
    [9] Hurwitz, A., “Sur quelques applications géométriques des séries de Fourier”, Ann. Sci. Éc. Norm. Supér. (3), 19 (1902), 357–408  crossref  mathscinet  zmath
    [10] Sachs, H., “Über eine Klasse isoperimetrischer Probleme: 1, 2”, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe, 8 (1958/59), 121–126, 127–134  mathscinet  zmath
    [11] Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia Math. Appl., 61, Cambridge Univ. Press, Cambridge, 1996, xii+329 pp.  mathscinet  zmath
    [12] Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, Cambridge Univ. Press, Cambridge, 1952, 324 pp.  mathscinet  zmath
    [13] Betta, M. F., Brock, F., Mercaldo, A., and Posteraro, M. R., “A Weighted Isoperimetric Inequality and Applications to Symmetrization”, J. Inequal. Appl., 4:3 (1999), 215–240  mathscinet  zmath
    [14] Blaschke, W., Kreis und Kugel, 2nd ed., De Gruyter, Berlin, 2013, 175 pp.  mathscinet
    [15] Henrot, A., Philippin, G. A., and Safoui, A., “Some Isoperimetric Inequalities with Application to the Stekloff Problem”, J. Convex Anal., 15:3 (2008), 581–592  mathscinet  zmath

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