Geodesic
On Vibrations of a Heavy Material Point in a Fixed Ellipsoidal Bowl
Russian journal of nonlinear dynamics, Tome 20 (2024) no. 4, pp. 449-461.

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The periodic motions of a material point are studied on the assumption that, throughout the motion, the point remains on a fixed absolutely smooth surface (in an ellipsoidal bowl), which is part of the surface of a triaxial ellipsoid. The motion occurs in a uniform field of gravity, and the largest axis of the ellipsoid is directed along the vertical. Cases are considered where the motion of the point occurs along one of the principal sections of the surface in the neighborhood of a stable equilibrium at the lowest point of the bowl. An analytical representation of the corresponding periodic motions is obtained up to terms of degree five inclusive with respect to the magnitude of perturbation of the point from the equilibrium. The stability of these periodic motions is investigated.
Keywords: nonlinear oscillations, normal forms, canonical transformations, stability 
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A. P. Markeev. On Vibrations of a Heavy Material Point in a Fixed Ellipsoidal Bowl. Russian journal of nonlinear dynamics, Tome 20 (2024) no. 4, pp. 449-461. http://geodesic.mathdoc.fr/item/ND_2024_20_4_a1/

[1] Uspekhi Mat. Nauk, 18:6(114) (1963), 91–192 (Russian) | DOI | MR | Zbl

[2] Arnol'd, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., 3, 3rd ed., Springer, Berlin, 2006, xiv, 518 pp. | DOI | MR | Zbl

[3] Moser, J. K., Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., 81, AMS, Providence, R.I., 1968, 60 pp. | MR

[4] Birkhoff, G. D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., 9, AMS, Providence, R.I., 1966, 305 pp. | MR

[5] Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., 8, Springer, New York, 1972, 369 pp. | DOI | MR | Zbl

[6] Malkin, I. G., Theory of Stability of Motion, U.S. Atomic Energy Commission, Washington, D.C., 1952, 456 pp.

[7] Malkin, I. G., Some Problems in the Theory of Nonlinear Oscillations, In 2 Vols.: Vol. 1, U.S. Atomic Energy Commission, Technical Information Service, Germantown, Md., 1959, 589 pp. | MR

[8] Guibout, V. and Scheeres, D. J., “Stability of Surface Motion on a Rotating Ellipsoid”, Celestial Mech. Dynam. Astronom., 87:3 (2003), 263–290 | DOI | MR | Zbl

[9] Koiller, J., Castilho, C., and Regis Rodrigues, A., “Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability”, Regul. Chaotic Dyn., 24:1 (2019), 61–79 | DOI | MR | Zbl

[10] Makhdum, B. and Nadim, A., “Dynamics and Equilibria of $N$ Point Charges on a 2D Ellipse or a 3D Ellipsoid”, Appl. Math., 14:4 (2023), 245–264 | DOI | MR

[11] Markeev, A. P., “On the Problem of Nonlinear Oscillations of a Conservative System in the Absence of Resonance”, Prikl. Mat. Mekh., 88:3 (2024), 347–358 (Russian)

[12] Gantmacher, F. R., Lectures in Analytical Mechanics, Mir, Moscow, 1975, 264 pp.

[13] Pars, L. A., A Treatise on Analytical Mechanics, Heinemann, London, 1965, 641 pp. | MR

[14] Markeev, A. P., Theoretical Mechanics, RCD, Institute of Computer Science, Izhevsk, 2007, 592 pp. (Russian) | MR