The Newtonian force experienced by a point mass near a finite cylindrical source

Abstract

The Newtonian gravitational force experienced by a point mass located at some external point from a thick-walled, hollow and uniform finite circular cylindrical body was recently solved by Lockerbie, Veryaskin and Xu (1993 Class. Quantum Grav. 10 2419). Their method of attack relied on the introduction of the circular cylindrical free-space Green function representation for the inverse distance which appears in the formulation of the Newtonian potential function. This ultimately leads Lockerbie et al to a final expression for the Newtonian potential function which is expressed as a double summation of even-ordered Legendre polynomials. However, the kernel of the cylindrical free-space Green function which is represented by an infinite integral of the product of two Bessel functions and a decaying exponential can be analytically evaluated in terms of a toroidal function. This leads to a simplification in the mathematical analysis developed by Lockerbie et al. Also, each term in the infinite series solution for the Newtonian potential function can be expressed in closed form in terms of elementary functions. The authors develop the Newtonian potential function by employing toroidal functions of zeroth order or Legendre functions of half-integral degree, Q m− 1 2 (β) (Bouwkamp and de

Generalized Heine’s identity for complex Fourier series of binomials

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“…Algebraic functions of the form (z − cos j) −m arise naturally in classical physics through the theory of fundamental solutions of Laplace's equation, and they represent powers of distance between two points in a Euclidean space. Their expansions in Fourier series have a rich history, appearing in the theory of arbitrarily shaped charge distributions in electrostatics (Barlow 2003;Popsueva et al 2007;Pustovitov 2008a,b;Verdú et al 2008), magnetostatics (Selvaggi et al 2008b;Beleggia et al 2009), quantum direct and exchange Coulomb interactions (Cohl et al 2001;Enriquez & Blum 2005;Gattobigio et al 2005;Poddar & Deb 2007;Bagheri & Ebrahimi 2008), Newtonian gravity (Fromang 2005;Huré 2005;Huré & Pierens 2005;Chan et al 2006;Ou 2006;Saha & Jog 2006;Boley & Durisen 2008;Mellon & Li 2008;Selvaggi et al 2008a;Even & Tohline 2009;Schachar et al 2009), the Laplace coefficients of the planetary disturbing function (D'Eliseo 1989(D'Eliseo , 2007, and potential fluid flow around actuator discs (Hough & Ordway 1965;Breslin & Andersen 1994), just to name a few physical applications. A precise Fourier analysis is extremely useful to fully describe the general non-axisymmetric nature of these problems.…”

Closed form expressions for gravitational multipole moments of elementary solids

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Magnetostatics of the uniformly polarized torus

“…The advance involves solving Poisson's equation with the particular inhomogeneous Dirichlet-type boundary conditions, dictated by the induced surface magnetic charges. A most straightforward way for applying the general integral form of solution turns out to be the use of the required Green's function in terms of toroidal functions (Cohl et al 2000); recently, Selvaggi (2005Selvaggi ( , 2008 has revived the interest in these functions by their successful and systematic implementation to the solution of magnetic, electrostatic and gravitational problems of potential theory. The stray magnetic field then follows in §2c through, essentially, straightforward differentiation in toroidal coordinates.…”