The “Phantom” Force

Abstract

THE famous principles of conservation and dissipation of energy, which have done so much to promote the progress of physical science in recent years, were undoubtedly first inferred and generalised from certain similar laws in the theory of forces which, as we find noticed by Prof. Tait in NATURE (vol. xiv. p. 462), were first propounded by Newton.¹ If in any mechanical system, Newton observes, urged by any forces, to which we must add those which arise from friction, the action of a force reckoned as a gain in the system is measured by the product of its impulse and the space through which it is pushed back, or as a loss in the system when the product relates to a space through which the force is allowed to act, and if as action of the same kind in the system we also count its gains and losses of actual energy of motion, the whole amount of action in the system remains unchanged during the motion. Viewed from the standpoint of the laws of motion, force, and matter, which Newton starts with in the “Principia,” and keeping in mind the special definition here given (coinciding with the modern term “potential increase”) of the “action” of a force, obviously the reverse of what would vulgarly be called the action of a force in increasing a body's energy of motion, this proposition at first looks like a truism; but the idea of potential energy here coined by Newton² is really an essential one; and it besides allows the mode of action of some forces of very common occurrence in nature to be described more simply than they could be without it. The force of gravitation, of attraction and repulsion between two bodies permanently electrified or magnetised, and all dual forces or actions and reactions directed along, and depending only on the distance between two bodies, and not at all upon the time, are of this kind. The force can be completely described in these cases (and it may be looked upon in the first instance as only a measure of convenience) by the permanent gradient of energy-variation everywhere; and hence also by the permanent change of energy from one distance to another, when, as is supposed in this example, the dual force pair acts along the line of centres; since then the changes of actual energy which it produces (acting alone upon the bodies) are independent of the rotation of this line, and may be regarded either as produced with the natural motion of this line's rotation or by the same forces acting along a fixed line of centres. When two such bodies approach, or recede from each other, whatever time elapses or whatever course they may pursue about their centre of mass, not only are the momentary transfers between actual and potential energy equal in energy value at every moment of the motion (for this is general, and by this condition only the bodies returning twice to the same distance from each other might have very different energies of motion at the two returns); but the whole energy of motion which can be gained between two distances is a definite one, and as this would not be so if the bodies returned twice to the same distance with different actual energies, nor if they returned twice to the same distance with different potential energies, it follows at once that not only is the sum of the actual and potential energies at any one distance invariable with the lapse of time and with any intervening motions of the bodies, but since the gain of actual energy from this distance to any other is the loss of potential energy, the sum of these two energies is also the same at one distance as it is at another, and it therefore varies neither with the time nor with the distance of the bodies from each other.