Mechanical Theories in
Mathematical Physics
Mechanical Theories in Mathematical Physics
By
Arthur Korn in Berlin.
Given the large number of new experimental facts that have emerged in recent decades, mechanics, with its endeavor to explain all
physical and chemical phenomena as movements of a (preferably unified) matter, has not always been able to keep up. New, sometimes very fantastic hypotheses have emerged, which often proved to be fruitful, even though they did not always align with our intuition. Without
denying the benefits that some of these new hypotheses, especially the theory of relativity and quantum theory, have brought, I have always opposed the extreme views that mechanical theories should now generally be dismissed as unproductive. I still believe that the
greatest progress will always come from mechanical theories, and it has repeatedly been shown that even those modern theories, which believed they had emancipated themselves the most from mechanical ideas, only advanced when a clearly mechanical idea was
introduced into them.
The first major attack on mechanical theories was delivered with the argument that mechanical theories are not capable of explaining the
phenomena of the electromagnetic field, as they are summarized by the Maxwell-Hertz equations. In fact, the mechanical explanation of the electromagnetic field is hardly feasible if one strictly adheres to the causal principle of classical mechanics (e.g., in the form of
D'Alembert's principle); however, it is by no means forbidden to supplement the hypotheses of mechanics so that certain new approximation terms are included in the fundamental equation of the principle when the velocities or accelerations are significantly greater
than those generally assumed in classical mechanics. It should not be surprising that the explanation of electromagnetic phenomena requires extraordinarily rapid oscillations with enormous frequencies, which did not occur in earlier mechanics, and therefore the
emergence of new approximation terms that can be neglected at usual velocities and accelerations should not be surprising either.
The first and most important task for a mechanical theory of the electromagnetic field is to answer the question: What mechanical
conception should we form of a (positive or negative) electric particle? Without delving into mathematical details here, I may say that there is no other possibility of mechanical explanation than the conception of electric particles as pulsating bodies, i.e., an electric particle is a
material particle that periodically changes its volume with an enormously high frequency. If the medium in which such pulsating particles are located is assumed to be incompressible or nearly incompressible for oscillations of very high frequency, mechanical interactions
between the particles arise (first studied by C. A. Bjerknes), similar to Coulomb forces between electric particles, if one distinguishes positive and negative electric particles by their phase (the positive particles reach the maximum volume when the negative particles have
the minimum volume, and vice versa). However, a difficulty initially remained with this mechanical conception of electric particles: Calculations based on classical mechanics result in attraction between particles with the same phase and repulsion between particles with
opposite phases; for a theory of Coulomb forces, the signs would have to be reversed. I overcame this difficulty in the following way: In the theory according to classical mechanics, the amplitudes of the pulsations of two particles, which have given mean kinetic energy in some
initial state, change when the geometric arrangement of the particles changes (albeit very slightly, by amounts that are small compared to the dimensions of the particles relative to their distances). If one now adds from the outset the condition that the pulsation amplitudes must
remain exactly constant, then the reversal of the sign for the mechanical interactions results. The hypothesis of the conservation of natural oscillations, which proves necessary here, cannot easily be justified using classical mechanics; it must be assumed that the matter forming
the electrons has the property of maintaining the mean energy of natural oscillations more accurately, the greater the frequencies of the natural oscillations are. For frequencies, as they occur in ordinary mechanics, nothing changes in the assumptions of classical mechanics,
but a new term must be considered in the fundamental principle of mechanics, which, at very high frequencies, includes this conservation of natural oscillations. The formula that includes this principle, I have called the principle of individuality, not as opposed to the principle of
relativity—my investigations go back to a time before the birth of the theory of relativity. The need for such an extension of classical mechanics is indicated by very general natural phenomena: the conservation of chemical atoms, cells in biology, etc. The principle of
individuality can be formulated as follows:


If we imagine that to the usual velocities
u0,v0,w0u0,v0,w0 are added oscillation velocities of very high frequency so that the actual velocities become
uu being the density of a small volume element of matter with an internal point (x, y, z) at time tt, δx,δy,δzδx,δy,δz
the virtual displacements of x,y,zx,y,z, then D'Alembert's principle should be modified as follows: (1)
where (ξ,η,ζ)(ξ,η,ζ) represents any point on the surface ∂ν∂ν of δνδν, δwδw is the outward normal of the surface element δwδw at the point (ξ,η,ζ)(ξ,η,ζ).
The equation obviously goes over into the usual D'Alembert's principle when One can bring this principle closer to intuition by considering a total reflection of the quantities
on the surfaces ∂ν∂ν, which only applies to oscillations of very high frequency. One might later demand the transition from D'Alembert's principle to this principle when the frequency of oscillations is initially thought to be lower (as in ordinary mechanics) and then transitions to
oscillations of higher and higher frequency. I would like to leave this development to the future and, in a sense, consider the equations (1) (D'Alembert's principle) and (2) (fundamental principle considering the conservation of individuality) as approximations on either side.
I would also like to emphasize that I assume the individuality correction only for the matter composing the electrons, not for the rest of the matter. However, I do not think of two qualitatively different materials, but rather assume that transitions between the two materials are also
possible. We have, in a sense, only two approximations here, and it should be left to future developments to find the transitions from the approximation equation (1) to the approximation equation (2).
Pulsating material particles, for which the individuality correction does not apply, exert mechanical interactions on each other, which are exactly opposite in sign to the mechanical interactions of electric particles; pulsating material particles of the same phase attract each other
according to Newton's law. According to my theory of "universal oscillations," a system of compressible particles embedded in an infinite, empirically incompressible matter is capable of a series of natural oscillations, and the fundamental oscillation is a pulsation of the
compressible particles with the same oscillation duration and phase. This fundamental oscillation should be the mechanical cause of gravitation, the compressible particles being the "gravitating particles."
In every atom, there are gravitating, positively electric, and negatively electric particles, which exert gravitational and Coulomb effects externally; within the atom itself, the interactions at small distances may obey entirely different laws. From the existence of chemical atoms,
each composed of a certain number aa of gravitating particles, a certain number bb of positively electric particles, and a certain number cc of negatively electric particles, certain conclusions can be drawn, but I will not go into them here. I would like to emphasize again that,
contrary to the currently widely prevailing relativistic explanations of gravitation, I also provide a mechanical explanation for gravitation through pulsating "gravitating particles," which, however, do not include the individuality correction in the dynamic fundamental law, unlike
the electric particles. The electric particles are therefore distinguished from the actual carrying mass by the conservation of their individuality; we can, in a sense, consider the electric particles as apparently living, and the gravitating particles as dead matter because
only the former retain their natural oscillations, their individuality, while this is not the case for the latter, and, for example, the gravitating particles change their pulsation amplitudes when the mutual geometric arrangement of the particles changes. For this reason, I have also
predicted that gravity on Earth must be different when the Earth is in aphelion than when it is in perihelion. It is possible that the differences are small, but perhaps they can also be detected through precise measurements.
Based on the principle of individuality, the entire electromagnetic theory, in the form of the Maxwell-Hertz equations, can be developed with the intuitive assumption that the electromagnetic field is characterized by velocities
(3) where TT represents an extraordinarily short oscillation duration, the Maxwell-Hertz electric components, and the Maxwell-Hertz magnetic
components are proportional, while represent the so-called visible velocities.
I had to add only one more hypothesis, which is not of a fundamental nature, but merely presupposes the existence of a certain initial state at some initial time, namely, that every translational velocity of an electron is associated with a rotation about an axis parallel to the velocity
direction, the relations (4)
where ωω is a function of x,y,z,tx,y,z,t and kk is a universal constant. Assuming that these equations are fulfilled at some initial time, they hold for all subsequent times, so that here no fundamental hypothesis is required, only the assumption of a particular initial state. This
principle of universal rotation, which I expressed about 20 years ago, has some affinity with Dirac's recent investigations of the "electron spin," although it does not entirely coincide formally.
If a mechanical theory of gravitation and the electromagnetic field could be built on these mechanical foundations, new investigations would be necessary for phenomena that are most frequently described today with the help of wave mechanics and quantum mechanics,


particularly the theory of spectra and atomic theory in general, and especially the theory of de Broglie waves and electron diffraction.
To penetrate these phenomena with mechanical theories, I will once again start from the hydrodynamic equations:
(5)
where pp represents pressure, ρρ the density, and u,v,wu,v,w the velocities for the point (x,y,z)(x,y,z) at time tt. These equations must be supplemented by an equation of state, i.e., a relation between pp and ρρ, which is illustrated by the kinetic theory of gases as the Mariotte
Gay-Lussac law (6)
where TT represents absolute temperature and RR the so-called gas constant; absolute temperature is set proportional to the mean kinetic energy of the disordered gas motion. The considerations of kinetic gas theory are only valid under the condition that the velocities
u,v,wu,v,w are small compared to the mean absolute disordered velocities that are added to u,v,wu,v,w, and that we are not dealing with oscillations of extraordinarily high frequencies; in the latter case, one can understand that the law (6) will undergo a modification. However,
we will assume the equation of state in the form: (7)
where we reserve the right to make further assumptions about the constant cc.
To simplify the situation, we will assume that the motion possesses a velocity potential, i.e.,
(8) and that forces of the type
act on each particle ρdνρdν, so that a force function ΨΨ is present: (9)
where ΨΨ is explicitly independent of tt. The hydrodynamic equations then reduce to a single one: (10)
The fundamental idea of my mechanical theory of wave mechanics is now that, in addition to the translational velocities which are determined by the Hamilton-Jacobi differential equation:
(11) boundary and initial conditions, oscillation velocities may be added in a new approximation, so that:
(12) then a linear equation for XX results:
(13) and if we set
(14) then the differential equation for the functions Ψj(x,y,z)Ψj(x,y,z) becomes:
(15) For simplicity, we will assume only one of the possible oscillations with frequency ν0ν0, so that the solution:
(16) where w0Ψ0w0Ψ0 satisfies the differential equation:
(17) If the constant cc of the equation of state is proportional to ν0ν0:
(18) then we obtain:
(19) thus arriving at the Schrödinger equation of wave mechanics. If the force function acting is a constant, then we have:
(20) where w0ν0w0ν0 represents the constant initial velocity. In this case (the case of de Broglie waves), we obtain a wavelength for the satellite
oscillations (21)
which is inversely proportional to the velocity ν0ν0, as experience demands.
In the case of a general force function ΨΨ, we obtain the satellite oscillations with the help of the equation
(22) as in wave mechanics; but it should now be noted from the mechanical point of view:
The force function is arbitrary with respect to an additive constant for given force components, which is only determined by initial conditions concerning (11); thus, for a particle in an atom with some initial velocity ν0ν0, a different series of natural oscillations will result for each
ν0ν0; but the differences will give a very specific series that is assigned to the given force components. With arbitrary initial velocities, the natural oscillations added
to the translational velocities would cover the entire spectrum, but the combination oscillations corresponding to the differences in frequencies would be regarded as the spectral oscillations.


Now the question arises: How is it that with the presence of a large number of oscillation series that fill the entire spectrum, oscillations with
frequency differences arise—a question whose answer also brings with it the explanation of the Raman and Compton effects. This question would be impossible to answer if one only operated with linear differential equations in theoretical physics, as was done for a long time.
However, if one also considers second-order terms in the equations of mathematical physics, as has already been done in the hydrodynamic equations, then combination oscillations always arise during the superposition of oscillations, which result from the addition
and subtraction of frequencies.
In the frequency differences
we have the preferred series that should be used to explain the spectra of atoms.
In one point, we have deviated from classical mechanics by assuming in the equation of state:
(23) that the constant cc is proportional to the frequency νν when the frequency is very high:
(24) whereas for low frequencies,
(25) according to the Mariotte-Gay-Lussac law, should be assumed. When several high frequencies are present simultaneously, assumption
(24) will have to be replaced by a somewhat more complicated assumption, which will not be discussed here.
How can we justify such an assumption (24) mechanically? Consider some particle of the medium in question, which is bombarded from all
sides; the pressure will obviously be proportional to the density, then to the mean absolute velocity of the impacting particles, and finally to the frequency of collisions. In kinetic gas theory, the frequency of collisions is also proportional to the mean absolute velocity of the
impacting particles, but under the assumption that the velocities u,v,wu,v,w are relatively small compared to these and that we are not dealing with oscillations of very high frequency. In the latter case, the frequency of collisions may satisfy a more complicated law, so that
the proportionality with the mean absolute velocity of the impacting particles gives way to proportionality with the frequency when this becomes very high. We then arrive at an equation of state
(26) in which
(27) thus becomes proportional to the frequency and the square root of the absolute temperature. We thus arrive at a mechanical justification
precisely for the equation of state we need for the mechanical theory of wave mechanics. The equation of state (26), (27) can also be usefully employed to explain photoelectric phenomena.