Translational transformations of tensor solutions of the Helmholtz equation and their application to describe interactions in force fields of various physical nature
Authors:
Yuri M. Urman and Sergey I. Kuznetsov
Journal:
Quart. Appl. Math. 72 (2014), 120
MSC (2000):
Primary 20G45, 33C55, 35C10, 35J05, 43A90
Published electronically:
November 19, 2013
MathSciNet review:
3185129
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References 
Similar Articles 
Additional Information
Abstract: Using group theory and irreducible tensor formalism we derive formulas for translational transformations of the tensor solutions of the Helmholtz equation. These formulas can be used to solve different problems in theoretical and mathematical physics, where it is necessary to relate boundary conditions for two or more spatial bodies. We show that these formulas can be used to perform invariant expansions of the interaction energy of the bodies in force fields of different physical nature. These expansions have a number of advantages and are very efficient and convenient to study force interactions. Examples from celestial mechanics, space vehicle dynamics and electric current interactions are given.
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Harper Langston, Leslie
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Willard
Miller Jr., Symmetry and separation of variables,
AddisonWesley Publishing Co., Reading, Mass.LondonAmsterdam, 1977. With
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M. Urman, Invariant expansion of the force function of mutual
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A. Orlov, An approximate representation of the potential of mutual
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Y. M. Urman, ``Application of the method of irreducible tensors to celestial mechanics problems,'' Astronomy Reports, vol. 39, pp. 531538, 1995.
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V. K. Abalakin, et al., Handbook on Celestial Mechanics and Astrodynamics. Moscow: Nauka, 1976.
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V. V. Beletskiy and A. A. Hentov, Rotary motions of a magnetized satellite. Moscow: Nauka, 1985.
 [1]
 Bernard Friedman and Joy Russek, Addition theorems for spherical waves, Quart. Appl. Math. 12 (1954), 1323. MR 0060649 (15,702h)
 [2]
 Seymour Stein, Addition theorems for spherical wave functions, Quart. Appl. Math. 19 (1961), 1524. MR 0120407 (22 #11161)
 [3]
 Orval R. Cruzan, Translational addition theorems for spherical vector wave functions, Quart. Appl. Math. 20 (1962/1963), 3340. MR 0132851 (24 #A2687)
 [4]
 M. Danos and L. C. Maximon, Multipole matrix elements of the translation operator, J. Mathematical Phys. 6 (1965), 766778. MR 0175515 (30 #5699)
 [5]
 M. N. Jones, ``A GroupTheoretical Formulation of Geophysical Elastodynamics,'' Proc. R. Soc. Lond. A, vol. 356, pp. 549568, 1977.
 [6]
 F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, An addition theorem for vector Helmholtz harmonics, J. Math. Phys. 21 (1980), no. 12, 27542755. MR 597591 (81m:33004), http://dx.doi.org/10.1063/1.524394
 [7]
 B. U. Felderhof and R. B. Jones, Addition theorems for spherical wave solutions of the vector Helmholtz equation, J. Math. Phys. 28 (1987), no. 4, 836839. MR 880310 (88m:33018), http://dx.doi.org/10.1063/1.527572
 [8]
 D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum theory of angular momentum, World Scientific Publishing Co. Inc., Teaneck, NJ, 1988. Irreducible tensors, spherical harmonics, vector coupling coefficients, symbols; Translated from the Russian. MR 1022665 (90j:81062)
 [9]
 E. Joachim Weniger and E. Otto Steinborn, A simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics, J. Math. Phys. 26 (1985), no. 4, 664670. MR 785677 (86f:33010), http://dx.doi.org/10.1063/1.526604
 [10]
 E. J. Weniger, ``Addition Theorems as 3D Taylor Expansions,'' International Journal of Quantum Chemistry, vol. 76, pp. 280295, 2000.
 [11]
 Y. M. Urman, ``Addition theorems for tensor spherical wave functions,'' Zh. Tekhn. Fiz., vol. 51, pp. 457462, 1981.
 [12]
 Ronald C. Wittmann, Spherical wave operators and the translation formulas, IEEE Trans. Antennas and Propagation 36 (1988), no. 8, 10781087. MR 962131 (89j:78027), http://dx.doi.org/10.1109/8.7220
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 W. C. Chew, ``A derivation of the vector addition theorem,'' Microwave and Optical Technology Letters, vol. 3, pp. 256260, 1990.
 [14]
 Daniel W. Mackowski, Analysis of radiative scattering for multiple sphere configurations, Proc. Roy. Soc. London Ser. A 433 (1991), no. 1889, 599614. MR 1116968 (92h:78012), http://dx.doi.org/10.1098/rspa.1991.0066
 [15]
 W. C. Chew and Y. M. Wang, ``Efficient ways to compute the vector addition theorem,'' J. Electromagn. Waves Appl., vol. 7, pp. 661665, 1993.
 [16]
 K. T. Kim, ``The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields,'' IEEE Trans. Antennas Propagat., vol. 44, pp. 14821487, 1996.
 [17]
 Kristopher T. Kim, Efficient recursive generation of the scalar spherical multipole translation matrix, IEEE Trans. Antennas and Propagation 55 (2007), no. 12, 34843494. MR 2433218, http://dx.doi.org/10.1109/TAP.2007.910358
 [18]
 K. T. Kim, ``Symmetry relations of the translation coefficients of the scalar and vector spherical multipole fields,'' Progress In Electromagnetics Research B, vol. 48, pp. 4566, 2004.
 [19]
 V. Rokhlin, ``Diagonal Form of Translation Operators for the Helmholtz Equation in Three Dimensions, Research Report YALEU/DCS/RR894, Dept. of Comp. Sci.,'' Yale University, New Haven, CT, 1992.
 [20]
 Michael A. Epton and Benjamin Dembart, Multipole translation theory for the threedimensional Laplace and Helmholtz equations, SIAM J. Sci. Comput. 16 (1995), no. 4, 865897. MR 1335895 (96f:35023), http://dx.doi.org/10.1137/0916051
 [21]
 W. C. Chew, Vector addition theorem and its diagonalization, Commun. Comput. Phys. 3 (2008), no. 2, 330341. MR 2389804 (2009c:78014)
 [22]
 B. He and W. C. Chew, Diagonalizations of vector and tensor addition theorems, Commun. Comput. Phys. 4 (2008), no. 4, 797819. MR 2463190 (2009j:43010)
 [23]
 T. J. Dufva, et al., ``Unified derivation of the translational addition theorems for the spherical scalar and vector wave functions,'' Progress In Electromagnetics Research B, vol. 4, pp. 7999, 2008.
 [24]
 W. Z. Yan, et al., ``On the Convergency Properties of Translational Addition Theorems,'' Progress In Electromagnetics Research Symposium, Beijing, China, March 2327, 2009.
 [25]
 B. He and W. C. Chew, ``The Tensor Addition Theorem: From the Viewpoint of Group Theory,'' presented at the Antennas and Propagation Society International Symposium, 2008. APS 2008. IEEE San Diego, CA, 2008.
 [26]
 Victor Twersky, Multiple scattering by arbitrary configurations in three dimensions, J. Mathematical Phys. 3 (1962), 8391. MR 0142299 (25 #5692)
 [27]
 E. A. Ivanov, Twobody Diffraction of Electromagnetic Waves [in Russian]. Minsk: Nauika I Tekhnika, 1968.
 [28]
 F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni, Use of group theory for the description of electromagnetic scattering from molecular systems, J. Opt. Soc. Amer. A 1 (1984), no. 2, 183191. MR 736077 (85b:78021), http://dx.doi.org/10.1364/JOSAA.1.000183
 [29]
 F. Borghese, et al., Scattering from model nonspherical particles, Springer, 2007.
 [30]
 B. He and W. C. Chew, ``Addition theorem'', in Modeling and computations in electromagnetics, Lect. Notes Comput. Sci. Eng., vol. 59, Springer, Berlin, 2008, pp. 203226. MR 2766831 (2012e:78017), http://dx.doi.org/10.1007/9783540737780_8
 [31]
 P. A. Martin, Multiple scattering, Encyclopedia of Mathematics and its Applications, vol. 107, Cambridge University Press, Cambridge, 2006. Interaction of timeharmonic waves with obstacles. MR 2259988 (2007k:35348)
 [32]
 N. A. Gumerov and R. Duraiswami, Fast multipole methods for the helmholtz equation in three dimensions, Elsevier, 2004.
 [33]
 N. A. Gumerov and R. Duraiswami, ``Fast, Exact, and Stable Computation of Multipole Translation and Rotation Coefficients for the 3D Helmholtz Equation,'' UMIACS TR, vol. 44, 2001.
 [34]
 L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325348. MR 918448 (88k:82007), http://dx.doi.org/10.1016/00219991(87)901409
 [35]
 Leslie Greengard and Vladimir Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 229269. MR 1489257 (99c:65012), http://dx.doi.org/10.1017/S0962492900002725
 [36]
 S. K. Veerapaneni, et al., ``A fast algorithm for simulating vesicle flows in three dimensions,'' 2011.
 [37]
 I. Lashuk, et al., ``A massively parallel adaptive fastmultipole method on heterogeneous architectures,'' in Proceedings of the Conference on High Performance Computing Networking Storage and Analysis SC 09 2009.
 [38]
 M. Harper Langston, Leslie Greengard, and Denis Zorin, A freespace adaptive FMMbased PDE solver in three dimensions, Commun. Appl. Math. Comput. Sci. 6 (2011), no. 1, 79122. MR 2836694 (2012k:65165), http://dx.doi.org/10.2140/camcos.2011.6.79
 [39]
 J. B. Minster, ``Transformation of multipolar source fields under a change of reference frame,`` Geophys. J. R. Astr. Soc., vol. 47, pp. 397409, 1976.
 [40]
 Y. M. Urman, ``Irreducible Tensors and Their Application in Problems of Dynamics of solids'', Mechanics of solids, vol. 42, pp. 5268, 2007.
 [41]
 Willard Miller Jr., Symmetry and separation of variables, AddisonWesley Publishing Co., Reading, Mass.LondonAmsterdam, 1977. With a foreword by Richard Askey; Encyclopedia of Mathematics and its Applications, Vol. 4. MR 0460751 (57 #744)
 [42]
 Yu. M. Urman, Invariant expansion of the force function of mutual attraction for a system of bodies, Astronom. Zh. 66 (1989), no. 5, 10811092 (Russian); English transl., Soviet Astronom. 33 (1989), no. 5, 556561 (1990). MR 1053690 (91c:70026)
 [43]
 A. A. Orlov, An approximate representation of the potential of mutual attraction between two bodies, Vestnik Moskov. Univ. Ser. III Fiz. Astronom. 1960 (1960), no. 3, 6976 (Russian). MR 0124523 (23 #A1835)
 [44]
 ``NIMA Technical Report TR8350.2,'' Department of Defense World Geodetic System, 1984.
 [45]
 (2011). ESA news: Earth's gravity revealed in unprecedented detail Available: http://www.esa.int/esaCP/SEM1AK6UPLG_index_0.html
 [46]
 Y. M. Urman, ``Application of the method of irreducible tensors to celestial mechanics problems,'' Astronomy Reports, vol. 39, pp. 531538, 1995.
 [47]
 V. K. Abalakin, et al., Handbook on Celestial Mechanics and Astrodynamics. Moscow: Nauka, 1976.
 [48]
 V. V. Beletskiy and A. A. Hentov, Rotary motions of a magnetized satellite. Moscow: Nauka, 1985.
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Additional Information
Yuri M. Urman
Affiliation:
Nizhny Novgorod State Pedagogical University, Nizhny Novgorod, Russia, and Nizhny Novgorod Institute of Management and Business, Nizhny Novgorod, Russia
Email:
urman37@mail.ru
Sergey I. Kuznetsov
Affiliation:
Johns Hopkins University, Baltimore, MD, USA, and Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
Email:
sergkuznet@hotmail.com
DOI:
http://dx.doi.org/10.1090/S0033569X2013013260
PII:
S 0033569X(2013)013260
Received by editor(s):
December 7, 2011
Received by editor(s) in revised form:
March 14, 2012
Published electronically:
November 19, 2013
Article copyright:
© Copyright 2013
Brown University
