# Force-producing models

# Force-producing models

# Abstract and Keywords

The correctness of Heinrich Hertz's image of mechanics is essentially reduced to one question: whether it is possible to construct hidden systems and connections to the tangible systems, such that the total system will obey the fundamental law, or said differently, such that the effect on the tangible system will mimic the forces empirically found in nature. Very few defenders of Hertz's approach to mechanics tried to support his image of mechanics by constructing concealed motions that would account for concrete forces. One serious attempt was made independently and more directly in 1916 by Franz Xaver Paulus, who treated conservative systems with monocyclic hidden masses. Paulus proposed force-producing models that may lend logical support to Hertz's image, but it is also evident that very few physicists would embrace an image of nature based on such artificial mechanisms. It is also rather obvious that it was not such models Hertz had hoped for.

*Keywords:*
force-producing models, forces, Franz Xaver Paulus, mechanics, connections, image, conservative systems, tangible systems, hidden systems

The correctness of Hertz’s image of mechanics is essentially reduced to one question. Is it possible to construct hidden systems and connections to the tangible systems, such that the total system will obey the fundamental law, or said differently, such that the effect on the tangible system will mimic the forces empirically found in nature. If such systems can be constructed Hertz’s image is correct; if it can be shown that such systems cannot be constructed the image will be incorrect. It is perhaps not surprising that critics of Hertz’s mechanics did not provide a proof that a construction is impossible. Indeed it is not so obvious how such an impossibility proof should be made, and, moreover, such a proof is only necessary if one accepts Hertz’s philosophical analysis. A critic of Hertz’s mechanics was more likely to reject at least parts of his philosophical analysis as well.

It is more surprising that very few defenders of Hertz’s approach to mechanics tried to support his image of mechanics by constructing concealed motions that would account for concrete forces. The only serious attempts were made by Brill (Brill 1909, pp. 28–30) (see Chapter 27) and independently and more directly in 1916 by a relatively unknown assistant at the Technische Hochschule in Vienna, Franz Xaver Paulus (1895–1949).

Paulus began his paper (Paulus 1916) with an admirably clear exposition of the main ideas in Hertz’s account of forces^{1}. Then he treated conservative systems whose hidden masses are monocyclic. Exactly like R. Liouville (see Section 18.6), to whom he did not refer, he remarked that if a monocyclic hidden system adds an amount to the kinetic energy of the system, the additional apparent potential energy would be *C* ^{2} *U* This is exactly the step from eqn (18.45) to eqn (18.48) if we assume that the constant value of is *C* instead of one. He then (following Boltzmann (Boltzmann 1891)) investigated a series of variations of the centrifugal regulator of which the following was the most general instance: Consider (Fig. 26.1) a system of one visible mass *m* that can move along a vertical *z*-axis and a hidden mass that can rotate around the *z*-axis in variable height and with variable distance from the *z*-axis. Assume that the two masses are coupled with a weightless string of fixed length that can roll over a weightless wheel *ω* that is forced through a weightless
(p.275)

*z*denotes the height of

*m*the coupling can be expressed by the equation

*z*= 0.

Assume, moreover, that is forced to move along a curve that rotates around the *z*-axis together with Then eqn (26.1) will take the form

*x*and the angle φ around the

*z*-axis as generalized coordinates of and expressed the kinetic energy of the hidden system as

According to the remark above this corresponds to *m* being subject to a potential energy

*m*to be subject to a given conservative force with a positive (decreasing) potential energy

*U*(

*z*). This can be obtained by a suitable choice of

*ƒ*. Indeed, from eqn (26.5) we see that

*z*is subject to gravitation we have

*U*(

*z*).I write ‘almost any’ because, as Paulus pointed out, the right-hand side of eqn (26.5) is strictly positive so the above mechanism can only produce the potential energy

*U*(

*z*) if

*U*is everywhere positive.

This is an immediate result of Hertz’s assumption that potential energy is, in fact, kinetic energy of a hidden system, kinetic energy being positive by definition. To be sure, one can always add or subtract an arbitrary constant from the function *U* so that one can reproduce negative potentials as well, but only if they remain bounded from below. For example, in the description of gravitation *k* in eqn (26.8) may be chosen arbitrarily large, but once it has been chosen *z* cannot obtain values less than *γ*/*k*.

Paulus only worked thoroughly through this one example of one point moving along a straight line, and merely indicated in a few words how he imagined one could generalize the mechanisms to more parameters and more cyclic coordinates (Paulus 1916, pp. 858–859).

More than 50 years later the leading MIT economist Paul Anthony Samuelson, independently of Paulus, investigated how Hertz’s mechanics could account for the motion of a point mass in a constant force field. He came up with a similar answer, although phrased in abstract analytical terms, without any mechanism to do the trick. Samuelson also pointed to the problem that *U* cannot become negative, and therefore *z* has a lower bound. He took that to be a strong argument against Hertz’s approach:

We can always select our arbitrary origin for altitude

zin such a way as to make the newgzof whatever sign we like at any range of altitudes. [This corresponds to an addition of an arbitrary constant to the potential energy]. But every time we consider a path that falls out of that range, we would have to change in advance our origin forz, a procedure both practically messy and aesthetically repugnant. Or, as a Hertzian would put it, this construction would seem to lack ‘appropriateness.’…Moreover, the phenomenon discussed here is general:

V(q)[U(z)] will for many natural problems want to run through a gamut of values from −∞ to +∞, leading to the same messy (p.277) requirement that we add new arbitrary constants toVin each different range. Why take a local train involving many transfers when an express train rides right through? (Samuelson 1971)

Samuelson’s argument assumes as a given thing that in nature there are potentials that are not bounded from below. However, there are good physical arguments to the contrary. In nature no (approximately) constant force field stretches infinitely far, and gravitational forces cease to be applicable when the gravitating bodies touch each other. The unboundedness only occurs because one makes the unphysical assumption that the attracting bodies are mathematical points. Thus, one may argue that Hertz’s assumption about bounded potential energy is in conformity with nature.

The force-producing models suggested by Paulus, may lend logical support to Hertz’s image, but it is also evident that very few physicists would embrace an image of nature based on such artificial mechanisms. It is also rather obvious that it was not such models Hertz had hoped for. Paulus’s mechanisms appear ad hoc, and each interaction will require its own set of mechanisms. What Hertz had in mind was a rather simple model of the ether, which could ‘explain’ all known interactions in one stroke. One might perhaps even have hoped that such a model would predict new phenomena, which Paulus’s mechanisms could not. After Einstein’s introduction of relativity theory interest in the ether gradually faded away, and with it interest in constructing better hidden force producing models to support Hertz’s mechanics.

## Notes:

^{1} He did not use Hertz’s geometric language, though he had himself independently suggested a similar language (Paulus 1910), nor did he treat non-holonomic constraints).