Potential Theory: the Origin and Applications

Potential theory: the origin and applications Irina Markina1 University of Bergen, Norway Abstract The potential theory comes from mathematical physics, in particular, from electrostatic and gravitational problems and finds their applications in the probability theory, scattering theory, biological systems and many other branches of science. During the last century the classical potential theory and the non-linear potential theory has occupied an important place in mathematics. We present the original gravitational and electrostatic problems that lead to the notion of capacity and harmonic functions. We give the short survey of potential theory and present some modern fields of its applications. Keywords: Potential theory, harmonic functions, stochastic process. 1. INTRODUCTION AND HISTORY The term “potential theory” arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace’s equation. Hence, potential theory was the study of functions that could serve as potentials. Nowadays, we know that nature is more complicated - the equations that describe forces are systems of non-linear partial differential equations, such as the Einstein equations, the Yang-Mills equations, and that the Laplace equation is only valid as a limiting case. Nevertheless, the term “potential theory” has remained as a convenient term for describing the study of functions satisfying the Laplace equation and its generalisations. As it was mentioned ,originally studies were related to the properties of forces which follow the law of gravitation. In the statement of this law given by I. Newton (1643 - 1727) in 1687 the only forces considered were the forces of mutual attraction acting upon two material particles of small size or two material points. These forces are directly proportional to the product of the masses of these particles and inversely proportional to the square of the distance between them. Thus, the first and the most important problem from the point of view of celestial mechanics and geodesy was to study the 1Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway. e-mail: [email protected] forces of attraction of a material point by a finite smooth material body - a spheroid and, in particular, en ellipsoid (since many celestial bodies have this shape). After first partial achievements by Newton and others, studies carried out by J. L. Lagrange (1736 - 1813) in 1773, A. Legendre (1752 - 1833) between 1784 - 1794 and by P. S. Laplace (1749 - 1827) continued in 1782 - 1799 became of major importance. Lagrange established that the field of gravitational forces, as it is called now, is potential field. He introduced a function which was called in 1828 by G. Green (1793 - 1841) a potential function and later in 1840 by C. F. Gauss (1777 - 1855) - just a potential. At present, the achievements of this initial period are included in courses on celestial mechanics. Already Gauss and his contemporaries discovered that the method of potentials can be applied not only to solve problems in the theory of gravitation but, in general, to solve a wide range of problems in mathematical physics, in particular, in electrostatics and magnetism. In this connection, potential became to be considered not only for the physically realistic problems concerning mutual attraction of positive masses, but also for problems with “masses” of arbitrary sign, or charges. Important boundary value problems were defined, such as the Dirichlet problem and the Neumann problem, the electrostatic problem of the static distribution of charges on conductors or the Robin problem, and the problem of sweeping-out mass (balayage method). To solve the above- mentioned problems in the case of domains with sufficiently smooth boundaries certain types of potentials turned out to be efficient, i. e., special classes of parameter - depen- dent integrals such as volume potentials of distributed mass, single - and double - layer potentials, logarithmic potentials, Green potentials, etc. Results obtained by A. M Lya- punov (1857 - 1918) and V. A. Steklov (1864 - 1926) at the end of 19th century were fundamental for the creation of strong methods of solution of the main boundary value problems. Studies in the potential theory concerning properties of different potentials have acquired an independent significance. In the first half of the 20th century, a great stimulus for the generalisation of the principal problems and the completion of the existing formulations in the potential theory was made on the basis of the general notions of a Radon measure, capacity and generalised functions. Modern potential theory is closely related in its development to the theory of analytic, harmonic and subharmonic functions and to the probability theory. Together with further studies of classical boundary value problems and inverse problems, the modern period of the development of potential theory is characterised by the applications of methods and notions of topology and functional analysis, and the use of abstract axiomatic methods. 2. SIMPLE EXAMPLES FROM GRAVITY AND ELECTROSTATICS While the theory of Newtonian potentials has various aspects, it is best introduced as a body of results on the properties of forces which are characterised by Newton’s Law of Universal Gravitation. Every particle of matter in the universe attracts every other particle, with a force whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other. If we denote by m1 the mass situated at a point P , by m2 the mass situated at a point Q, then the force is expressed in the following formula: −→ m m F = γ 1 2 , r2 where r is the distance between P and Q. The constant of proportionality γ depends solely on the units used. Newton’s law is stated in terms of particles. We usually have to deal, not with particles, but with continuously distributed matter. We then naturally think of dividing the body into small parts by the method of the integral calculus, adding the vector forces corresponding to the parts, and passing to the limit as the maximum diameter of the parts approaches 0. 1. Example of straight homogeneous segment. Let us consider a straight line segment, which we regard as having mass, so distributed that the mass on any interval is proportional to the length of the interval. The constant factor of proportionality λ is called the linear density. We have here an idealisation of a straight wire, which is the better approximation the smaller the diameter of the wire relatively to its length and the distance away of the attracted particle. Let axes be chosen so that the ends of the wire are the points (0, 0, 0) and (l, 0, 0). Let the attracted particle be in line with the wire, at (x, 0, 0), x > l, as in Figure 1. Let the wire be divided into intervals by the points 0 = ξ0, ξ1,ξ2,..., ξn = l. Then the interval (ξk,ξk+1) carries a mass λ △ ξk, which, by our physical law, is to be regarded as concentrated at some point ak of the interval. ak P 0 ξk ξk+1 l x Figure 1 The force, due to the particle thus concentrated, will lie along the x-axis, and will be given, in attraction units, by △ △ λ ξk △ △ Xk = − 2 , Yk =0, Zk =0. |x − ak| The force, due to the entire segment [0, l], will be the limit of the sum of the forces due to the system of particles or l λdξ λl X = − 2 = − , Y =0, Z =0. Z0 (x − ξ) x(x − l) The result may be given a more suggestive form by introducing the total mass M = λl, and by considering where in the segment a particle of that mass should be placed in order to yield the same attraction on a unit particle at P (x, 0, 0). If c is the distance from this point to P , then λl M X(P )= − = − and c = x(x − l). x(x − l) c2 p Thus the wire attracts a unit particle at P as if the mass of the wire were concentrated at a point of the wire whose distance from P is the geometric mean of the distances from P to the ends of the wire. As P approaches the nearest end of the wire, the force becomes infinite, but only like the inverse first power of the distance from this end to P , although a particle would produce a force which becames infinite like the inverse square of the distance: X(P )= M − c2 . The difference is that in the case of the particle, P moves to the whole mass, whereas in the case of the wire the mass distributed over the segment is concentrated at one of whose points does P moves arbitrary close. As P recedes further and futher away, the equivalent particle (as we call the particle with the same mass as the wire, and with the same attraction on a unit particle at P ) moves toward the mid-point of the wire, and the attraction of the wire becomes more and more nearly that of a fixed particle at its mid-point. Indeed, if d is the coordinate of the equivalent particle, then l d = x − c = x − x(x − l) → as x →∞. 2 p The equivalent particle is situated inside of the wire for any position of P . l 2 Example. Let us put the unite mass particle at the point P ( 2 ) on the perpendicular bisector of the material segment as shown in Figure 2 l P ( 2 ,y) △ Fk r 0 ξk ξk+1 l Figure 2 The distance r to the attracted particle from a point (ak, 0, 0) of the interval (ξk,ξk+1) is given by l r2 = a − 2 + y2, k 2 and the magnitude of the force at P , due to a particle at this point, whose mass is that on the interval (ξk,ξk+1) is λ △ ξk λ △ ξk △ Fk = = .

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