An Historico-Critical Account of Potential Energy: Is PE Really Real?

The concept of quantity of matter, what would later become “mass,” was introduced in the 13th century by Aegidius Romanus, who applied the already well-accepted notion of conservation of matter to the problem of transubstantiation as it arises in the Eucharist. For more on the subject, see Max Jammer, Concepts of Mass (Dover Publications, Mineola, NY, 1997), p. 45.

There are those interpreters of the special theory of relativity who maintain that mass is a function of speed, although that seems increasingly to be a minority position. Above and beyond this speed dependence (which is not accepted herein), mass varies with rest energy and so the idea of mass as a fixed quantity is an anachronism. The mass of an object is certainly Lorentz invariant, but it can nonetheless change as its rest energy changes.

Classical momentum, mass times velocity, was often erroneously expressed as weight times velocity. And that confusion runs all the way up into the 19th century. See for example G.P. Quackenbos, Natural Philosophy (D. Appleton and Co., New York, 1859), p. 29.

The primitive medieval notion of conservation of matter (see Ref. 1) no doubt gave impetus to the 17th-century quest for a dynamical conservation principle.

Huygens seems to have been aware of the essence of the principle of conservation of kinetic energy as early as 1652; see Ref. 1, p. 63.

See, for example, E. Hecht, Physics: Algebra/Trig (Brooks/Cole, Pacific Grove, CA, 2003), p. 85, or E. Hecht, Physics: Calculus (Brooks/Cole, Pacific Grove, CA, 2000), p. 117. When it came to falling bodies, Huygens might well be considered a successor of Galileo. In the Horologium Oscillatorium (1673), he sets out the principle that “when any number of weights starts to fall, the common centre of gravity cannot rise to a height greater than that from which it started.” This hypothesis would have a strong effect on Daniel Bernoulli's fluid dynamics.

The word Kraft in German means force, power, vigor, or energy, and that too contributed to the general confusion in terminology for decades; vis viva, living force, wasn't force at all, it was the precursor of kinetic energy. However daunting the task, Newton was very careful to define force, and he did it in a way that still has validity. Nonetheless, the linguistic mess continued for a remarkably long time. In fact, 19th-century authors often equated momentum to force and “impact.” Thus, T. Cavallo, in his text The Elements of Natural or Experimental Philosophy (T. Dobson & Son, Philadelphia, 1819), p. 19, wrote, “The momentum is the force of the body in motion, and equivalent to the impression, it would make on another body at rest, that should be presented to it precisely in the direction of its motion.” And J.L. Comstock in A System of Natural Philosophy (Pratt, Oakley & Co., New York, 1857), p. 39, states that “this power, or force, is called the momentum of the moving body.”

See, for example, R. Dugas, A History of Mechanics (Editions du Griffon, Switzerland, 1955), p. 235, or Thomas Hankins, Science and the Enlightenment (Cambridge University Press, Cambridge, 1985), p. 30.

The word energy was still being used rather arbitrarily well into the 19th century. For example, W. Peck in his text Introductory Course of Natural Philosophy for the Use of Schools and Academies (A.S. Barnes & Co., New York, 1875), p. 25, wrote, “The intensity of a force is the energy with which it acts.”

See James C. Maxwell, Matter and Motion (Dover, New York, 1991), p. 54. Interestingly, there are several popular books in English (e.g., Ref. 3, p. 31) that speak about “striking force” instead of “living force.”

Ernst Mach (1872), in his History and Root of the Principle of the Conservation of Energy (Open Court Pub. Co., IL, 1910), p. 19, called it “the theorem of the conservation of work,” and later in 1883 in The Science of Mechanics (Open Court Pub. Co., IL, 1960), p. 600, he referred to it as the “principle of vis viva.” R.B. Lindsay in Student's Handbook of Elementary Physics (Dryden Press, New York, 1943), p. 23, calls it “the work-energy theorem.” Since the 1980s there have been a number of articles pointing out the failings that arise when applying this theorem to nonpoint masses. See A.

John

Mallinckrodt

and

Harvey S.

Leff

, “,” , (April ) for an extensive bibliography.

Daniel Bernoulli earlier used the term vis potentialis to contrast with vis viva. In the text Popular Physics (American Book Co., New York, 1888), p. 35, J.D. Steele says that “energy may be either active or latent.”

Among the earliest texts to incorporate the new vocabulary of energy was W.J. Rolfe and J.A. Gillet, Handbook of Natural Philosophy for School and Home Use (Potter, Ainsworth, and Co., New York, 1869). On page 275 in the appendix they distinguish between “actual or dynamical energy” and “possible or potential energy.” In 1882 these authors published Natural Philosophy for the Use of Schools and Academies (Potter, Ainsworth, and Co., New York), and this time a far more extensive discussion of kinetic and potential energy appears on p. 28.

Maxwell (Ref. 10) correctly states (p. 56) that “work, therefore, is a transfer of energy from one system to another,” but alas he had previously (p. 54) said that “Energy is the capacity of doing work” and so creates a tautology. The failure of that often-repeated definition of energy (especially as it relates to the second law of thermodynamics) has been discussed extensively in the contemporary literature. Though it is still doggedly recited in dictionaries and introductory textbooks, energy is not the ability to do work.

Thomas Preston, The Theory of Heat, 2nd ed. (Macmillan and Co., London, 1904), p. 90.

W. T.

Stace

of Princeton University, in “,” (), – (July 5, ), seems to think so: “It is only by the means of the fiction of ‘potential’ energy that it is possible to hold that the same amount of energy is always in existence. All that the evidence shows, all that is empirically verifiable, is that where there is a certain quantity of energy which suddenly disappears out of existence at a certain time, the same amount of energy will reappear in the universe at some later time. The gap between the two existences of the energy is filled up by the fictitious supposition that it goes on existing ‘potentially.’”

E.M. O' Connor, Potentiality and Energy: A Dissertation (The Catholic University of America Press, Washington, D.C., 1939), p. 37.

J.W.N. Sullivan, The Limitations of Science (Mentor Books, New York, 1949), p. 155 (first printed by Viking Press in 1933).

E.N. Hiebert, Historical Roots of the Principle of Conservation of Energy (State Historical Society of Wisconsin, Madison, 1962), p. 102.

W.G.V. Rosser, Introductory Special Relativity (Taylor & Francis, London, 1991), p. 163. For a bibliography on the issue, see

L.B.

Okun

, “,” , ().

Anna Beck, Collected Papers of Albert Einstein (Princeton University Press, Princeton, NJ, 1989), Vol. II, Doc. 47, p. 286. See also Leo Sartori, Understanding Relativity (University of California Press, Berkeley, CA, 1996), p. 206.

For a calculation showing that the mass of the Earth should be reduced by a multiplicative factor of about $4.2 × 10−10$ upon compacting into a sphere, see Julian Schwinger, Einstein's Legacy (Scientific American Books, New York, 1986), p. 136. Also look at

Bertram

Schwarzschild

, “,” , (November ).

Ralph

 

Baierlein

, “ ,”  , – (March ) and

Ralph

Baierlein

, “” , – (May ). The conceptual dichotomy alluded to above seems to arise out of the distinction between what we have taken as a free particle's invariant mass (m), and what was once often referred to as its speed-dependent “relativistic mass” $(mr).$ Thus, if we assume the mass of a particle to be invariant, a photon has zero mass. Alternatively, if mass is assumed (as it is by Baierlein) to be speed dependent, a photon, even though it has zero rest mass $(mo),$ has a nonzero relativistic mass $(mr = γmo = E/c2).$ That, in part, gives the appearance of validity to the notion that mass is never converted into energy. Just think of the decay of a neutral pion (of mass $mπ$⁠) into two gamma photons. The process can be understood from both perspectives. But the interpretation in which photons have zero mass demands that the pion's mass be converted into the combined energy of the two photons $(mπc2).$

Albert Einstein, Out of My Later Years (Philosophical Library, New York, 1950), pp. 49 and 119.

Albert Einstein and Leopold Infeld, The Evolution of Physics (Simon and Schuster, New York, 1938), p. 200.

A. Einstein, Philosopher-Scientist, edited by Paul Schilpp (Harper & Row, Publishers, New York, 1959), p. 61.

The ball's gravitational field travels outward from it at the speed of light and so the ball can only interact with a limited portion of the universe, albeit an ever-increasing one. Much of the universe doesn't even know the ball has come into existence.