Full Text
Toward the History of Russian Science
N. A. Umov’s Views on Potential Energy, on Forces Acting at a Distance, and on Mass
A. I. Bachinsky.
Half a century ago N. A. Umov (then a 27-year-old docent) published three works1 devoted to the development of the following ideas.
By virtue of the law of conservation of energy, the total quantity of energy in an isolated system does not change, but energy may undergo transformations. Thus, kinetic energy may be transformed into potential energy. Kinetic energy is a measure of the intensity of a real phenomenon—motion. Umov asks: what real phenomenon corresponds to potential energy?
To this question science gives only one answer: potential energy represents the intensity of the work of forces. But can work be that real phenomenon which increases at the expense of the decrease of the system’s living force and, so to speak, survives this living force?—No, because the idea of work is inevitably connected with the idea of a current change. One can speak of the work of a force only so long as the point of application of the force is in motion. As soon as it comes to a state of rest, there is no more work.
The only exact and clear answer to the question posed consists, in Umov’s opinion, in the following assumption: potential energy is nothing other than the living force of the motions of certain media, imperceptible to us.
From this point of view Umov formulates the law of conservation of energy in the following way:
“a) Every change in the magnitude of living force is conditioned by its transition from the particles of one medium to the particles of other media, or else from some forms of motion to others.
b) A definite quantity of living force remains equal to itself under every change of phenomena.
c) The quantity of the living forces of nature is unchanging.”[^2]
Then Umov turns to the interpretation of the phenomena of the seeming actio in distans, or of the interaction of various agents (ponderable matter, electric charges, magnetic poles, electric currents) at distances. In all these phenomena there figures potential energy or (according to Umov’s view) the kinetic energy of intermediate media not subject to our observation (such as, for example, the ether). Umov shows how, with the aid of certain simple assumptions concerning the motion of the particles of these intermediate media, it becomes possible to arrive at the generally known laws of interaction of the above-mentioned agents.
Among the papers of N. A. Umov there has been preserved a copy of a letter written by him to some physicist who objected to the views developed by Umov in the articles mentioned. In the letter Umov defends his views, and, several times, while pointing to the connection of these views with the theories of Western scholars, uses expressions of an obviously ironic character. Evidently N. A. Umov’s correspondent belonged to that widespread type of person in Russia who accepts as truth nothing that is not supplied with a Western label. Unfortunately, the name of the correspondent cannot be established: the copy in our possession contains neither an address nor any indications whatsoever relating to the personality of the addressee.
The letter is of special interest because a whole series of propositions, of fundamental importance for characterizing Umov’s views, is developed here more fully and carried further than was done by Umov in his printed articles. For this reason it seemed appropriate to us to publish this letter as a valuable document from the history of Russian science.
In particular, one may emphasize the complete coincidence of two of Umov’s views with the views of Hertz, which the latter set forth two decades after Umov in his mechanics. Namely:
a) potential energy is in reality the kinetic energy of hidden masses (Hertz, Prinzipien der Mechanik, § 605).
b) the definition of mass (ibid. § 4).
It is known that N. A. Umov warmly welcomed Hertz’s mechanical theories and propagated them in some of his popular-science articles1.
Letter of N. A. Umov to an Unknown Physicist
When two inelastic bodies strike one another, everyone knows that, among other things, heating of the bodies results. Suppose that one of them was at rest. Would you really say that the amount of heat developed by the impact, expressed in mechanical units, is equal to the vis viva lost by the inelastic body that struck it? And you would be quite right. It follows from this that, in the case of collision of inelastic bodies as well, the sum of the vis viva of their particles remains the same—heat is motion, as it was before the impact. After this I do not understand in what way you write that experience shows that vis viva is lost? One must add that only the vis viva of a certain kind of motion is lost, but it can always be found in motion of another kind. If mechanics teaches that only the quantity of motion is conserved, it is because it looks only at the motions of visible bodies, and also because it is “rational” mechanics, and not the mechanics of nature. This example, which you point out to me as showing the inadequacy of my theory—because no matter how much you rack your brain, you cannot explain the loss of vis viva in the impact of inelastic bodies—, as you see, presents a case where attachment to rational mechanics leads to errors, since here too, as I have indicated, the vis viva of all motions (including heat) remains unchanged...*)
I now pass to other objections, and in doing so I shall base myself on rational mechanics and shall support my conclusions by references to Kirchhoff’s works, because I do not have one of Helmholtz’s articles at hand, while Maxwell’s article “On the electromagnetic field” I am only just reading with the help of an Englishman specially invited for this purpose.
It seems to me that we can indeed accept as unrestrictedly valid for nature the law of conservation of energy:
\[ T+\Pi=\mathrm{const.}, \]
where \(T\) is vis viva, and \(\Pi\) is potential energy. Let us imagine two bodies (I shall speak of mass later), surrounded by some medium; for simplicity I shall assume that this medium is an incompressible fluid. I make this assumption because the properties of the medium between us have not yet been under dispute. Let \(T_1, T_2\) be the vis viva of both bodies, and \(T\) the vis viva of all particles of the incompressible fluid surrounding the bodies. If I suppose that no external forces act on the bodies, and consequently that they themselves do not act on one another at a distance, then I must put \(\Pi=0\), and then I find \(T=\mathrm{const.}\), or
\[ T_1+T_2+T=\mathrm{const.} \tag{1} \]
From this I conclude that the change in the vis viva of the bodies is equal—in opposite sign—to the change in the vis viva of the particles of the intervening medium. This conclusion is inevitable for any theory that does not admit action at a distance. If action at a distance is admitted, the law of conservation of vis viva must be written as follows:
\[ T_1+T_2+\Pi=\mathrm{const.} \tag{2} \]
Comparing (1) and (2), I find:
\[ \Pi=T+\mathrm{const.}, \tag{3} \]
or, with a known choice of the constant:
\[ \Pi=T. \tag{3 bis} \]
This means that the potential energy is equal to the sum of the vis viva of the intervening medium. This conclusion and the consequences that follow from it constitute my fault. By God, it was not I—
*) Here one sentence has been omitted which, adding little to the substance of the matter, makes the exposition obscure. A. B.
first to do it; I merely said what others had not finished saying, but what they implied for particular cases. I generalized it and derived further consequences. Thus, I set to proving that I had stolen my idea from the works of the famous scholar Kirchhoff. I suppose that you have nothing against his authority (one could prove that Helmholtz, too, has my pernicious little ideas, and I think also Maxwell).
The well-known scholar Kirchhoff says the following in his article “Über die Kräfte zweier Ringe in einer Flüssigkeit”1: if there exists, in an incompressible fluid surrounding two rings, a velocity potential satisfying certain conditions, then, denoting by \(P\) the potential of two currents flowing around the rings, one upon the other, and by \(T\) the vis viva of all the particles of the fluid, we have (Crelle Bd. 71, p. 273):
\[ T+P=\mathrm{const.} \tag{4} \]
This equation is a particular case of equation (3). It coincides with it under a certain choice of constants. Kirchhoff merely did not notice that equation (4) holds in general in a theory reflecting interaction at a distance. Consequently, equation (3), with the fundamental idea implied in it, was used by Kirchhoff; I borrowed it from him, and not only it, but even stole the expression for the quantity \(T\), i.e. the vis viva of the particles of the intermediate medium. From the analysis of the law of equality of action and reaction I arrive at the expression:
\[ T=k\iiint\left[\left\{\frac{dV_1}{dx}+\frac{dV_2}{dx}\right\}^{2} +\left\{\frac{dV_1}{dy}+\frac{dV_2}{dy}\right\}^{2} +\left\{\frac{dV_1}{dz}+\frac{dV_2}{dz}\right\}^{2}\right]\,d\omega, \tag{5} \]
where \(V_1\) and \(V_2\) are the velocity potentials caused in the given case by the two rings. Kirchhoff, denoting by \(U_1, U_2\) the potentials of the rings at a magnetic pole, finds (p. 271):
\[ T=\frac{\rho}{2}\iiint d\omega\left[ \left\{\frac{dU_1}{dx}+\frac{dU_2}{dx}\right\}^{2} +\left\{\frac{dU_1}{dy}+\frac{dU_2}{dy}\right\}^{2} +\left\{\frac{dU_1}{dz}+\frac{dU_2}{dz}\right\}^{2}\right]. \tag{6} \]
Thus, I borrowed even the most essential part of my derivation from the authoritative scholar Kirchhoff, making it only more general: without connecting it with any properties of the medium, I derived it from the analysis of the equality of action and reaction.
It remains for me to speak about force and mass. Let Kirchhoff himself tell you about force. If \(\varphi\) is the velocity potential, and if it does not explicitly contain time, then:
\[ \frac{p}{\rho}=-\frac{1}{2}\left\{ \left(\frac{d\varphi}{dx}\right)^2+ \left(\frac{d\varphi}{dy}\right)^2+ \left(\frac{d\varphi}{dz}\right)^2 \right\}: \]
Therefore, in giving expression (4), Kirchhoff says: “\(\delta T\) ist das Moment der Druckkräfte,” etc. (the rest is clear). But I see that you are still dissatisfied, and therefore I shall try to explain the meaning of force in my theory otherwise.
Let us imagine three bodies \(A, B, C\) (which, for simplicity, I represent to myself as three points) in equilibrium. Equilibrium is possible in the case when, for a given position of the bodies, the vis viva of the intermediate medium is a maximum or a minimum, i.e. when \(\delta T=0\). Denoting by \(x_1, y_1, z_1\) the coordinates of the point \(A\), I find:
\[ \frac{dT}{dx_1}\delta x_1+\frac{dT}{dy_1}\delta y_1+\frac{dT}{dz_1}\delta z_1=0 \tag{8} \]
for all \(\delta x_1,\ \delta y_1,\ \delta z_1\); hence:
\[ \frac{dT}{dx_1}=0;\quad \frac{dT}{dy_1}=0;\quad \frac{dT}{dz_1}=0. \tag{9} \]
If by \(T_{bc},\ T_{ab},\ T_{ac}\) I denote the living forces of the intermediate medium, conditioned by the presence in it of the pairs of bodies \((BC),\ (AB),\ (AC)\), then from (9) it follows:
\[ \frac{dT_{ab}}{dx_1}+\frac{dT_{ac}}{dx_1}=0 \quad\text{and so forth.} \]
Consequently, that which must be destroyed—must be equilibrated—in order that the body \(A\) be in equilibrium is measured by the quantities:
\[ \left. \begin{array}{ccc} \dfrac{dT_{ab}}{dx_1}, & \dfrac{dT_{ab}}{dy_1}, & \dfrac{dT_{ab}}{dz_1},\\[6pt] \dfrac{dT_{ac}}{dx_1}, & \dfrac{dT_{ac}}{dy_1}, & \dfrac{dT_{ac}}{dz_1} \end{array} \right\}. \tag{10} \]
These quantities we call quantities of force. From this there follows, moreover, noting that reduced forces (as a very simple argument shows) “act” (fictionally) parallel to the coordinate axes:
1) the forces acting in nature are represented by derivatives of certain functions (this is not a repetition of the initial point, because at the beginning I supposed that there are no forces at a distance).
2) equation (8) is nothing other than the principle of virtual displacements in the case of nature.
Is it necessary to say that the equations of motion are obtained very simply as well? Even from them we can obtain a still better conception of force, considering force as the product of some coefficient \(m\) by acceleration, and taking as the unit of force the acceleration, i.e. the force acting on a body when its coefficient \(m=1\).
Namely, we can obtain the equations of motion from Hamilton’s principle, applying our notation:
\[ \delta \int (\Pi+T)\,dt=0. \tag{11} \]
If no forces at a distance act on our bodies, then \(\Pi=0\), and
\[ \delta \int (T_1+T_2+T)\,dt=0. \tag{12} \]
From this we shall arrive at equations of motion into which accelerations enter, and we shall find that the products of accelerations by certain specific coefficients \(m\), depending on the bodies, are represented in a known way through the expressions (10), whence we conclude that the latter represent forces acting in known directions.
From this it is already clear what I understand by mass. If this word “atom” I replace by the word “center of discontinuity,” or simply “a point at which the continuity of the medium spread through world-space is violated,” then the number of such centers, multiplied by a certain specific coefficient (if all the centers are homogeneous, i.e. consist of particles of one and the same substance, as we usually say), is mass. From the equations of motion, which you will obtain from (12), it is easy to convince oneself that mass, so defined, is the ratio of force to acceleration. Consequently, force has a real meaning to the same extent that it has a relation to acceleration. The latter, however, is caused by transformations of living force in the intermediate medium.
Now I go further. If you are consistent, then your view will pass from your editorial office into mine. Namely: you are a physicist and therefore know that when one phenomenon disappears, another must appear, equal to it in its intensity. A stone has lost living force by changing its position relative to the earth. This new
the position indicates that the matter has not remained in the same state as it was before—that changes have taken place. The changed position of the stone is a sign of the changes that have occurred in the phenomenon, and nothing more. Can one really seek the source of living force in a dead, geometrical difference of positions? Clearly not; geometry will not create motion for you; one must seek, therefore, a phenomenon which in its intensity would be equal to the lost living force. Where, then, is the vanished living force; where is the phenomenon corresponding to it? In the surrounding medium? You say—no. In the earth—it cannot be, for the stone gives up living force, for it comes into motion. It remains for you, therefore, only to admit énergie emmagasinée in the stone, if only you are consistent.
(Here the text of the copy ends.)