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), 1-20

MSC (2000):
Primary 20G45, 33C55, 35C10, 35J05, 43A90

Published electronically:
November 19, 2013

MathSciNet review:
3185129

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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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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/S0033-569X-2013-01326-0

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