Geodesic
Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 5-12 Cet article a éte moissonné depuis la source Math-Net.Ru

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When studying physical phenomena in spatial regions bounded by spherical or slightly non-spherical surfaces, spherical functions and solid spherical harmonics are widely used. A problem of transformation of those functions and harmonics with translation of the coordinate system frequently arises. Such a situation occurs, in particular, when the hydrodynamic interaction of spherical or slightly non-spherical gas bubbles in an unbounded volume of incompressible fluid is described. In the two-dimensional (axisymmetric) case, when the role of spherical functions is played by the Legendre polynomials, such a transformation can be performed using a well-known compact expression. Similar known expressions in the three-dimensional case are rather complex (they, for example, include the Clebsch–Gordan coefficients), which makes their application more complicated. The present paper contains derivation of such an expression, naturally leading to a compact form of its coefficients. Those coefficients are, in fact, a generalization to the three-dimensional case of the analogous known coefficients in the two-dimensional (axisymmetric) case.
Keywords: solid spherical harmonics, parallel translation. 
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A. A. Aganin; A. I. Davletshin. Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 5-12. http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a0/

[1] Lamb H., Hydrodynamics, Cambridge Univ. Press, Cambridge, 1916, 728 pp. | MR

[2] Fermi E., Notes on Quantum Mechanics, A Course Given by Enrico Fermi at the University of Chicago, Univ. of Chicago Press, Chicago, 1961, 171 pp. | MR | MR

[3] Duboshin G. N., Handbook for Celestial Mechanics and Astrodynamics, Nauka, Moscow, 1976, 864 pp. (In Russian)

[4] Idelson N. I., Potential Theory and Its Application to Geophysical Problems, GTTI, Moscow, 1932, 350 pp. (In Russian)

[5] Aganin A. A., Davletshin A. I., “Simulation of interaction of gas bubbles in a liquid with allowing for their small asphericity”, Mat. Model., 21:9 (2009), 89–98 (In Russian) | Zbl

[6] Aganin A. A., Guseva T. S., “Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields”, Fluid Dyn., 45:3 (2010), 343–354 | DOI | MR | Zbl

[7] Aganin A. A., Davletshin A. I., “Interaction of spherical bubbles with centers located on the same line”, Mat. Model., 25:12 (2013), 3–18 (In Russian) | MR | Zbl

[8] Aganin A. A., Davletshin A. I., Toporkov D. Yu., “Dynamics of a line of cavitation bubbles in an intense acoustic wave”, Vychisl. Tekhnol., 19:1 (2014), 3–19 (In Russian)

[9] Davletshin A. I., Khalitova T. F., “Equations of spatial .hydrodynamic interaction of weakly nonspherical gas bubbles in liquid in an acoustic field”, J. Phys.: Conf. Series, 669 (2016), Art. 012008, 4 pp. | DOI

[10] Aganin A. A., Davletshin A. I., “A refined model of spatial interaction of spherical gas bubbles”, Izv. Ufim. Nauchn. Tsentra Ross. Akad. Nauk, 2016, no. 4, 9–13 (In Russian)

[11] Doinikov A. A., “Mathematical model for collective bubble dynamics in strong ultrasound fields”, J. Acoust. Soc. Amer., 116:2 (2004), 821–827 | DOI

[12] Takahira H., Akamatsu T., Fujikawa S., “Dynamics of a cluster of bubbles in a liquid (theoretical analysis)”, JSME Int. J. Ser. B, 37:2 (1994), 297–305 | DOI

[13] Hobson E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge Univ. Press, Cambridge, 1931, 500 pp. | MR | Zbl

[14] Steinborn E. O., Ruedenberg K., “Rotation and translation of regular and irregular solid spherical harmonics”, Adv. Quantum Chem., 7 (1973), 1–81 | DOI