Once Again on Inertial Forces1

In Connection with the Article by A. N. Krylov2

L. I. Mandelstam

The difficulties that arise in the question of the so-called inertial forces are connected chiefly with two points: 1. Continuous bodies are considered, for example: a flywheel, a thread, etc. (and in this case the constraints are also realized by continuous bodies), while the terminology of “point” mechanics is applied. 2. The constraints are regarded as absolutely rigid.

In essence, of course, there are no inertial forces, neither real nor fictitious. However, in view of established custom, it is necessary also to say how this term can be understood harmlessly.

It must be noted that the following passage in Newton’s Principia is misleading: “Definition III. The innate force of matter is a power of resisting inherent in it, by which every body, so far as it is left to itself, maintains its state of rest or of uniform rectilinear motion. This force is always proportional to the mass, and differs from the inertia of the mass only in our manner of conceiving it.” “From the inertia of mass it comes about that every body is only with difficulty put out of its state of rest or motion; and therefore the innate force could very reasonably be called the force of inertia...”3

All this says little. As far as I remember, Maxwell, in Matter and Motion, subjected this passage to criticism. In any case, one cannot derive from it anything about the “reality” or “fictitiousness” of inertial forces. Apparently, however, the beginning of the whole dispute is this passage in Newton.

Such an “argument” about the reality of inertial forces, of course, cannot withstand criticism: the flywheel is torn apart, therefore inertial forces are real. With the same right one could say: “By pumping air in, we can burst a boiler. By heating the air—also. Hence, when we heat we add something to the boiler, that is, heat is a substance!”

I shall begin with the motion of a planet around the sun. A planet moving approximately in a circle is acted upon by the force of attraction of the sun. According to Newton’s third law, the sun is acted upon by an equal force of attraction of the planet, directed in the opposite direction. No other forces act here. One may call the first force in this case centripetal, and the second—centrifugal. But this is only another name for Newtonian forces of attraction, and not some new forces; and since we hold that the name does not affect the substance of the matter, the question of the “reality” of the centrifugal force is devoid of content.

Let us turn to a little ball moving in a circle and held by a string. We shall regard the little ball as a point of mass \(m\). Then, according to Newton, a force acts on it (of course, a real one, caused by the tension of the string), directed toward the center. It is equal to the mass multiplied by the acceleration: \(mj\). The argument then proceeds as follows: by Newton’s third law, action is equal to reaction; hence, a force \(mj\), directed away from the center, acts on the end of the string. This force is called the force of inertia. This is a complete misunderstanding. It is appropriate to reason in one of two ways: either 1) to regard the string as consisting of discrete material points, or 2) to regard the string as a continuous body.

Let us first examine the first of these methods and, for simplicity, take as a model of the string a one-dimensional chain of points (Fig. 1). \(O\) is a fixed center, \(m\) is a material point, and the remaining points \(1, 2,\ldots\) are the discrete material points of which the string consists. Since we take point mechanics as the basis of the model and choose the corresponding model, we must accept that the forces acting between neighboring points act “at a distance,” in the same sense as the sun acts on the planets. Then the case of a ball on a string differs from the case of a planet’s motion around the sun only in that here there are many points, and in the law according to which the force depends on the distance. Just as, in the motion of a planet around the sun, it is inexpedient and not customary to say that a centrifugal force acts on the sun from the side of the earth, so here there is no basis for saying that a centrifugal force acts on point 7 (Fig. 1). In any case, in principle (apart from the different dependence of the force on the distance), there is no difference between these two cases.

Fig. 1.

In the second way of considering the matter, we regard the links as continuous bodies. Here it is advisable not to distinguish between a material point and a string, but to take as the basis a continuous rod, as in Fig. 2. The case of a string with a ball is only a particular case of the type, say, shown in Fig. 3. With this way of considering the matter

Fig. 2.          Fig. 3.

one must introduce the concept of stresses in the body and proceed, in accordance with Newton’s third law, from the equality and oppositeness of the stresses acting in the section \(s\) on part \(I\) and part \(II\) of our body. The forces corresponding to these stresses are caused by the deformations of our body. These are real forces, in no way different from static forces under the same deformations. Only in motion are these deformations caused precisely by the circumstance that, in order to make the rod move in a circle, forces are needed which are conditioned by the deformations.

The origin of the deformations in the transition from rest to motion is perfectly obvious (as also in the discrete case). The individual points \(1, 2, \ldots, m\) would move, say, under the action of an impulse, along straight lines (tangents). In this motion there arise precisely those deformations which produce the required forces.

Here too one may, for the force acting from part \(I\) of the body on part \(II\), introduce the name centripetal, and for the force acting on part \(II\), centrifugal force. I believe that from what has been said above it is clear that centrifugal and centripetal forces are only another name for ordinary forces (elastic intermolecular forces or forces of attraction). It is also clear how pointless it is to discuss their reality or unreality. The very same forces which we always regard as forces, namely, as stated, elastic intermolecular forces and forces of attraction, sometimes begin to be called centrifugal.

One may raise the question whether such a designation is rational in general, or whether there are cases when it is rational. In any case, the term “centrifugal force,” especially in engineering, has taken root. Is there any basis for this? I think so, and here is why and when.

In those cases where—as in the case of a planet—it is required to find the motion under given forces, the name centripetal or centrifugal force is, at best, superfluous. But there are other cases, when bodies that are extremely little deformable take part in the motion; in other words, bodies in which the forces or stresses increase so rapidly with changes in intermolecular distances or deformations that we know in advance with sufficiently great accuracy [[unclear: continuation cut off at bottom of page]].

known characteristics of the motion. For example, although (from the physical point of view) there are no inextensible strings and absolutely rigid bodies, there are strings and solid bodies which, under such small extensions, give such large forces that we may regard the length of these strings as constant. Thus, for example, we know in advance that a ball tied to such a string will move in a circle. This is the basis for the enormous value of introducing rigid constraints in analytical mechanics. Suppose that, in the case of the motion of a ball on such a string, we are also given the tangential velocity. This specification and the practical inextensibility of the string determine the motion. At the same time, we do not know and have no need to know the deformations, and therefore we are deprived of the possibility of determining from the deformations the tensions in the string that are important to us.

In this case we determine these tensile forces by means of Newton’s second law from the accelerations, which we know, since we know the motion. In precisely these cases the designation, in essence, of ordinary elastic forces as centrifugal or centripetal forces may to a certain extent be justified. Here these names express the method of calculating them from the centrifugal or centripetal accelerations. But no more than this may be put into these names.

It is true that there is still one more justification for introducing the term “force of inertia,” even in such a case as the motion of planets. But this is something different. The point is that, when one wants to reduce a problem of dynamics to a problem of statics, then, since in statics the conditions of equilibrium are formulated for forces, in dynamics as well one tries to call certain expressions forces, in order to apply the conditions of statics literally. Namely, if I call \(-mj\) the force of inertia acting on the mass itself, and not, as is usually said, on the constraint, then the equations of motion are obtained from the following principle. Let us also call \(\pi = F - mj\), i.e., the sum of the acting force and the force of inertia, the lost force. Then, in order to find the equations of motion, one must express that the system of points, under the action of the lost forces applied to them, with the given constraints, is in equilibrium. Here the term “force of inertia” is simply the name of the quantity \(-mj\). This is something different from what we earlier called a force of inertia (\(\equiv\) centrifugal, centripetal force). Here it is not a force in the sense of Newton’s second law, but a name for the expression \(-mj\). A dispute about reality is meaningless here as well.

1. Prepared from a draft manuscript by M. A. Leontovich. ↩

2. A. N. Krylov, “On the Forces of Inertia and d’Alembert’s Principle” (in the book: A. N. Krylov, Thoughts and Materials on the Teaching of Mechanics. Publishing House of the Academy of Sciences of the USSR, Moscow—Leningrad, 1943). ↩

3. Translation by A. N. Krylov. Emphasis by L. I. Mandelstam. ↩