On Gerber’s Velocity-dependent Gravitational Potential
I have made a simple discovery about Gerber’s velocity-dependent potential that casts new light on the Gerber-Seeliger controversy as well as on the Lagrangian method in application to such potentials. In brief, toward the end of the nineteenth century Paul Gerber (thought by Beckmann  to be a German high-school teacher), claimed to deduce from the assumption of retarded action of gravitational potential a velocity-dependent “generalized potential” that predicted a force producing Mercury’s observed anomalous perihelion advance . This was many years before Einstein’s general theory predicted exactly the same thing on the basis of entirely different assumptions.
There is little doubt that Gerber’s assumption of speed-c retarded force action of gravity is in conflict with practically everything else known about gravity’s distant force action. (If there were such a retardation it would throw off the dynamics of the solar system in readily observable ways.) So, his physics can safely be ignored. However, as a semi-empirical exercise in formal mathematical modification of Newton’s gravitational potential, the type of investigation pioneered by Gerber is of continuing interest. This is particularly true since Seeliger, a German astronomer, in 1917 claimed  that Gerber got his mathematics all wrong – so there are mathematical points to be explored, related to the handling of velocity-dependent potentials. I have studied a number of authors on this subject and have found only one who has gone into it really extensively and carefully. This is Assis, and I intend to be guided here by his text , where issues of mathematical procedure arise. Some more famous authors, such as Whittaker , while not mathematically wrong, can be so misleading on the side of physics as to make little difference from being wrong.
The main point to be noted about the Lagrangian method is that when potential energy depends only on position of the particle described (I shall here simplify by considering a one-particle system) there is a direct relationship between the Lagrangian function L and physical quantities; namely, , where ^ is kinetic energy of the particle and V is the physical potential energy of that particle. But when potential energy depends on particle velocity as well as position the formalism loses direct contact with physics and requires a Lagrangian function employing a so-called “generalized potential energy” having nothing to do with, and quite distinct from, the physical potential energy. So V no longer enters L. The physical potential energy must then be replaced by some non-physical function which we shall designate as S (following Assis ) . Thus
where , the q’s being Lagrangian generalized coordinates and the ’s being the corresponding generalized velocities. This S-function (not to be confused with Hamilton’s principal function) must be contrived to “give the right answer” – meaning that the right answer (the forces or equations of motion) must be known from other considerations. The Hamiltonian H does not suffer from this detachment from physics. It is always equal to the total physical energy, , and is related to L by
Eq. (2) is often spoken of as the “definition” of H, but that is a mathematician’s way of looking at it. The physics is all in H, which always depends on the physical potential energy V; whereas L depends on the non-physical “generalized potential energy” S in the case of a velocity-dependent physical potential energy, .
Now suppose a very perverse thing. Suppose some clever fellow has done our contriving for us, and we know the function ^ . Is there any way we can deduce from this knowledge the physical potential energy V? The answer is yes, Eq. (2) will allow us to deduce it. I have not seen Gerber’s original paper , only Seeliger’s critique  of it … so there may be some historical distortion in what follows. As I now reconstruct matters, Gerber proposed that for gravity our function S (which he called V) be defined by
where r is a generalized Lagrangian coordinate describing the separation of a planet from its sun, , and is some mass parameter. The problem being non-relativistic, kinetic energy can be represented by . Defining L by Eq. (1), viz.,
Gerber then calculated the gravitational force acting on the planet via the equivalent of Lagrange’s equations,
Let us work through this calculation. We have from (4)
whence , (6a,b)
and . (6c)
Recognizing the term in (6b) as force (from Newton’s second law), and introducing (6) into (5), we have
This is the physical force obtained by Gerber. About it, Seeliger  says, “From this one sees, without any calculation, that there results indeed a motion of the perihelion of Mercury by an amount close to that empirically found for it; while, as is well-known, from Weber’s law only one-third of this amount results.” Weber’s law, in this context, postulates the physical potential energy to be
and the “generalized Lagrangian potential” to be
Note the important sign difference between these two. (Actually, Weber’s law concerned charged particle electrodynamics, but we here extend its application to gravity.) From Seeliger’s remark about the factor of three, we might hazard a guess that the proper expression for a physical potential energy to describe gravity [referring to (8a)] would be
Let us proceed to test this guess. We wish to discover the physical potential energy ^ consistent with Gerber’s proposed S, Eq. (3); that is, consistent with the Lagrangian (4). The information we want can be obtained from Eq. (2), which for our single Lagrangian coordinate takes the form
Using (6a) and (10), we find
from which, by inspection, we confirm our guess (9); namely,
We need by independent deduction to verify that this physical potential (12) leads to Gerber’s force law, Eq. (7). There are a number of ways to do this, several of which are described by Assis . By far the simplest is to recognize that the physical potential energy V is always related directly to observable force by a gradient operation. Thus in our one-dimensional case
By direct calculation from (12)
This requires no more than the recognition that both r and are continuous functions of t. If is imagined to be solved for , then ; so it is legitimate to differentiate with respect to r. (This was essentially Weber’s original argument and method .) Putting (14b) in (14a) yields
which is just the Gerber force law, Eq. (7), that we set out to verify. Hence we have established that ^ given by Eq. (12) [and guessed in Eq. (9)] is the physical potential energy required to ensure that the resulting non-Newtonian force [Eq. (7) or (15)] is just such as to predict the same perihelion advance of Mercury as Einstein’s general theory. (The algebraic sign of this force is not to be taken seriously. It can always be adjusted as need be. The signs used in the older literature for force and potential vary almost ad lib.) For translation of such calculations into multi-coordinate forms and vector notation, see Assis .
Concerning mathematical methods in mechanics, it is worth directing attention to the vast labor-saving in getting from (13) to (15) via the gradient of a physical quantity, as contrasted with the ponderous and tricky machinery of the Lagrangian method. The end result is the same – namely, a force law, (15), that apparently improves on Newtonian gravity, but agrees with Newton in the limit . This labor-saving is a direct consequence of working with a physically meaningful quantity, the physical potential energy, as contrasted with its “generalized” Lagrangian counterpart (which, in the case of velocity-dependent potentials, is in fact a generalization of nothing physical).
In summary, we know that Newton’s law of gravity, expressed by the potential energy
gives a remarkably accurate accounting of the dynamics of the solar system. It now seems, as a pure empiricism, that a slight modification of this physical potential energy to
where c is the speed of light, accounts quantitatively for the known departure from Newtonian predictions (the Mercury perihelion anomaly).
This physical potential energy is the only one a practical investigator really needs. But if mathematical complication is valued for its own sake, and the scholar consequently insists on a Lagrangian formulation, it is then necessary to use instead the “generalized potential” of Eq. (3); viz.,
This has the correct dimensions of potential energy, but has no direct connection with the physics and no role to play in either mathematics or physics except to correct the wrong answers that would be yielded by the Lagrangian method if the physical potential energy were mistakenly used instead. Of course, one could do every bit as well using the general theory of relativity – and this is certainly the course to be recommended to scientists who are reassured by complexity, or who feel insecure with approximations and need to lean on ultimate truth.
The foregoing, though simple enough, is a comparatively little-understood aspect of Lagrangian mechanics, which has led some great minds into gross error. Thus Seeliger , unless I am mistaken, fell into the trap of mistaking Gerber’s Lagrangian “generalized potential” S, Eq. (3), for a physical potential energy V. Hence he tried to use S in a gradient calculation of type (13)-(15), above, to derive the force. Naturally, he failed to confirm Gerber’s result and got only nonsense. This led him unjustly to blame Gerber for the nonsense. Why did Seeliger (and others including myself) fall into this trap? Perhaps because Gerber used the notation V (a standard form for physical potential energy) for what we have called S. It cannot be over-emphasized that simple gradient calculations of force require as operand the physical, not the Lagrangian “generalized,” potential energy.
Whittaker  does everything technically correctly, treating Weber’s problem in the Lagrangian formalism with [Eq. (8b)] instead of [Eq. (8a)]. But he uses the V notation for , and gives no warning to neophytes that this “generalized potential energy function” has nothing to do with a physical potential energy and thus is useless for anything but a Lagrangian calculation. Also, by strangely terming the Lagrangian a “kinetic potential,” and by writing it as , he confers on it an apparent physicality that may deceive not only his fellow mathematicians but a few unwary physicists as well. To repeat, all my warnings here refer to the case of velocity dependence of the physical potential energy. Where potentials depend only on position, the Lagrange formalism is both neat and trustworthy. But, where physical potentials are more complicated, it is neither.
To reiterate the main message of this paper: The physical potential energy law for gravity, embodied in Eq. (17), is new to me and possibly to the reader. I have not seen it in the literature. It is so simple and “cute” – and does the same job for Mercury’s perihelion as general relativity theory for so much less pain – that it might actually prove useful for very refined (yet practical) problems of astro-navigation, etc. Having published on the “Gerber-Seeliger controversy” before  in support of Seeliger, I must now eat crow.
 P. Beckmann, Einstein Plus Two (Golem Press, Boulder, CO, 1987), p. 171.
 P. Gerber, Zeit. f. Math. u. Phys. 43, 93-104 (1898).
 H. Seeliger, Ann. d. Phys. 53, 31-32 (1917).
 A. K. T. Assis, Weber’s Electrodynamics (Kluwer, Dordrecht, 1994).
 E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Dover, N.Y., 1944).
 T. E. Phipps, Jr., Galilean Electrodynamics 14, 22 (2003).
T. E. Phipps, Jr.
15 May, 2004
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