Notes on the Principle of Action and Reaction in General Dynamics
|Notes on the Principle of Action and Reaction in General Dynamics (1908)
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Bemerkungen zum Prinzip der Aktion und Reaktion in der allgemeinen Dynamik, Physikalische Zeitschrift 9 (23): 828-830,
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(Session from September 23, 1908.)
The Newtonian principle of equality of action and reaction is well known to consist mainly of the theorem of the constancy of momentum or impulse of motion; and I would therefore like to speak of such a principle only in the meaning of this theorem, especially in its importance for the general dynamics that not only covers mechanics in the narrower sense, but also electrodynamics and thermodynamics.
Many of us probably still remember the stir that was caused, when H.A. Lorentz in his construction of atomistic electrodynamics on the basis of a stationary aether, denied the universal validity of the third Newtonian axiom, and it was inevitable that this circumstance was considered (for example by H. Poincaré) as a serious objection to Lorentz's theory. A kind of stabilization only came back as it turned out, especially by the investigations of M. Abraham, that the reaction principle can still be saved in its full generality, if we introduce a new momentum in addition to the mechanical momentum only known so far, i.e. the electromagnetic momentum. Abraham has made this even more plausible, by bringing the conservation of momentum in comparison with the conservation of energy. Similarly, as the energy principle is violated when we don't consider the electromagnetic energy, and fulfilled when we introduce this type of energy, the reaction principle is violated if we only consider the mechanical momentum, and is satisfied on the other hand, as soon as we have considered the electromagnetic momentum.
However, this certainly indisputable comparison doesn't affect a still substantial difference. As to the energy we know already a number of different forms: the kinetic energy, gravitation, the elastic deformation energy, heat, chemical energy, and it is therefore not a principally new feature when to these different forms, the electromagnetic energy is attached as another form. Conversely, until now only a single form was known as momentum: mechanical momentum. While the energy from the very outset represented a universal physical expression, the momentum was so far a special mechanical expression, the reaction principle a special mechanical theorem, and therefore the extension recognized as necessary was perceived as a revolution of principle, by which the previous relatively simple and uniform concept of momentum becomes considerably more complicated in nature.
Is it not possible, even from the standpoint of general dynamics, to form the definition of momentum (although it now includes both the mechanical and the electromagnetic form) in the same uniform way as it was earlier done in mechanics? A positive answer to this question would certainly lead to an advance in the understanding of the true meaning of the reaction principle.
In fact, such a uniform definition of momentum seems to be possible and feasible, at least if we also admit of Einstein's theory of relativity. However, it must now be emphasized that this theory is regarded nowadays by no means assured. But as their deviations from other eligible theories are only limited to very small terms, one may say however, that they can be considered as correct just up to those differences, and to that extent, therefore, the following considerations keep in all circumstances a certain importance.
In the theory of relativity, the momentum can now be generally attributed to that vector which expresses the energy flux, not only the Poynting electromagnetic energy flux, but the energy flux in general. From the standpoint of direct action theory all types of energy can change their location within space only by continuous propagation, not by changes in the form of leaps. Therefore, the energy principle generally requires that the change of total energy contained in a certain space is like a surface integral, namely, the algebraic sum of the entire incoming energy through the surface of that space. The flux can be submitted by radiation, such as the Poynting vector, by conduction as with pressure or with pushes and with heat conduction, and by convection as with the admission of ponderable atoms or electrons through the considered surface. In any case, the entire energy flux at every point in space, with respect to unit surface and unit time, is a certain finite vector, and the ratio of this vector by the square of the speed of light c is in general the momentum in relation to unit volume.
Take for example a ponderable fluid moving with velocity q under the pressure p. Through a surface element df of a stationary area perpendicularly directed to q, the mechanical work is: p · df · qdt. The entrained energy is: df · ε · qdt, where ε denotes the energy density. Accordingly, by that definition the momentum of unit volume is:
Comparing this expression with the usual mechanical momentum kq, where k is the density of the liquid, we find:
a known relationship of the theory of relativity.
From the point of view described we can generally denote the principle of equality of action and reaction as the "theorem of inertia of energy".
But we can go a step further. As the constancy of energy implies the concept of energy flux, the constancy of momentum also implies the concept of "flux of the momentum", or briefly spoken: the "momentum flux". As the momentum located in a particular area can only change due to external effects, i.e. according to the theory of direct action only by processes at the surface of that space, the amount of change in unit time is a surface integral which can be described as the total momentum flux in the interior of that space. A major difference, however, to the energy flux is the fact that the energy is a scalar, yet the momentum is a vector. Therefore, the incoming energy in that space is expressed by a single surface integral, and the energy flux is a vector. However, the momentum flux into that space is expressed by three surface integrals, corresponding to the three components of momentum, and the momentum flux in one place is a tensor triple, in the notation of W. Voigt, which is characterized by six components
To get an idea of the significance of these tensor triples, we first consider the mechanical momentum and its corresponding mechanical momentum flux. The total momentum flux into the interior of a room, i.e. the increase in momentum in the interior per unit time, is equal to the resulting mechanical force which acts on the entire mass located within that space. Consequently, the momentum flux through a surface element is nothing else than the mechanical pressure on the surface element, and its components have the form:
where n denotes the inner normal of the surface element. are the six components of the tensor triples representing the momentum flux.
The same is true for the electromagnetic momentum flux in a vacuum. The components of this tensor triple are nothing else than the known Maxwell stresses. Its integration over a closed surface gives the total momentum flux into the interior, and thus the increase of all mechanical and electromagnetic momentum contained within the enclosed space. It is remarkable, how by this theorem the physical meaning of the Maxwell stresses is increased for the stationary aether theory. Because, as a force of pressure these stresses make not really sense in this theory, since we may not attribute a meaning to a force acting on something absolutely immovable. The fact that the Maxwell stresses, although they were officially abolished, have maintained themselves within the theory of a stationary aether, because they often proved so convenient as mathematical tools for certain calculations, could already suggest the idea, that they play some special physical role by which they are legitimate also for the stationary aether.
It is tempting to transfer the concept of momentum flux to the gravitational field, where, apart from the fatal sign, there are given a remarkable number of analogies; but a more detailed discussion of this problem would take too far at this place.
Minkowski: The theorems on momentum can be derived directly from the energy theorem in my point of view. Namely, the energy equation in Lorentz's theory depends on the reference frame for space and time. If we write the energy theorem for each possible reference frame, we have some equations, and in those the theorems on momentum are included.
Planck: Certainly. But I consider the independence from the reference frame not as a solid physical result, but more as a hypothesis, which I consider as promising but not to be established yet. However, we just need to consider whether these relationships actually exist in nature. This we can only learn by experiment, and hopefully the time is not far off when we experience it.
- See especially F. Hasenöhrl (Sitzungsbericht de. Akad. d. Wiss. zu Wien vom 31. Oktober 1907, S. 1400), although he doesn't directly proceed from the theory of relativity, but as far as I can see he get quite the same results.
- See for example M. Planck, Ann. d. Phys. (4), 25, 27, 1908. equation (48).
- See M. Abraham, Enzyklopädie d. math. Wiss. IV, 14, p. 28
- Vgl. H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen, p. 28. Leiden 1895.
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