Potential,

Potential, in dynamical science, is a quantity of peculiar importance. Its value, as a mathematical function in the theory of attraction, was recognised by Laplace in the Mécanique Céleste. The name was, however, given by George Green (1793–1841) in 1828, when its broad dynamical significance was for the first time explicitly stated and powerfully developed. The theory of the potential, in fact, is co-extensive with the dynamics of what are known as Conservative systems. When such a system is made to pass from one configuration to another, the work done against the forces of the system depends only upon the initial and final configurations, and in no way upon the particular series of changes by which the passage is made. For instance, the work done against gravity in lifting a given mass to a height of 500 feet is exactly the same whether the mass is lifted vertically up, by a balloon, say, or more laboriously taken up the gentle slope of a hill. The earth and the mass form, so far as gravitation is concerned, a conservative system. Practically, however, in dragging a mass up a slope a certain amount of work, greater or smaller according to circumstances, must be done against friction, and this will depend upon the character of the course taken. We know that the work so done is lost and cannot be recovered in dynamic form (see ENERGY). These forces are in short dissipative, and so far as their action is concerned the system is not conservative, and the theory of the potential does not apply. A little consideration will show that when the forces are functions of distances only the system will be conservative. Such forces then have a potential; and, although this does not exhaust all types of force-systems which have a potential, it includes all that are certainly known to occur in nature around us. The force of gravitation and the force between electrified or magnetised bodies evidently belong to the category just described. In all such cases the potential at any point in the field of force is a definite function of the position, a mathematical expression having for any particular case a definite value, such that the difference of the potentials of two points measures the work done in carrying unit quantity (of matter, electricity, magnetism, &c.) from the one point to the other (see ELECTRICITY for some further properties of the potential). If we take the two points very close to each other, we see at once that the small difference of the potentials must equal the product of the average force into the corresponding small distance. Thus, in the notation used in the article Calculus (q.v.), we have $\backslash Delta\; V\; =\; S\; \backslash Delta\; s$, where $V$ is the potential, $S$ the force, and $\backslash Delta\; s$ the small distance. Hence $S\; =\; dV/ds$ or the force in any direction is numerically equal to the rate of change of the potential per unit-length in that direction. When the potential is known a simple differentiation in any chosen direction gives the force in that direction. It is obvious that other directed quantities besides forces may be expressible as the differential coefficients of a single non-directed or scalar quantity. Thus, in the mathematical theory of Hydrodynamics (q.v.) a very important distinction is made between motions which have a velocity-potential and motions which have not. In the former the velocity can be represented as a space differentiation of a scalar quantity; in the latter it cannot. See VORTEX for an account of fluid motion, which has no velocity-potential.