What was meant by the 'ponderomotive force' as understood by Minkowski? Skimming through Minkowski's famous 1907 paper, he uses the term ponderomotive force.
What does he mean by this?
 A: Let's look at some clues as to what it probably meant at the time. The word is ponderomotive rather than pondermotive and is constructed like electromotive, magnetomotive, from ponder-o-motive. The [etymology][1] of ponder is given as

ponder early 14c., "to estimate the worth of, to appraise," from O.Fr. ponderare "to weigh, poise," from L. ponderare "to ponder, to consider," lit. "to weigh," from pondus (gen. ponderis) "weigh" (see pound (1)). Meaning "to weigh a matter mentally" is attested from late 14c.

Therefore as an initial guess, it could mean the line integral between two points of a force that acts upon substance to give it weight; perhaps the line integral of the Newtonian gravitational force?
Book Googling 'ponderomotive' turns up a quote from Energy and Empire: a biographical study of Lord Kelvin

what makes an electrified body move?
In May of 1843 Thomson published in the Cambridge Mathematical Journal a paper of a mere two pages which marks his earliest consideration of ponderomotive forces on electrified bodies. 'On the attractions of conducting and non-conducting electrified bodies' showed that, for a given distribution of electricity on the surface of a body A, the total moving force exerted on A by an arbitary electrical mass M is the same whether A be a conductor or non-conductor.

Hermann von Hermholtz and the foundations of nineteenth-centurey science by David Cahan

For he sought to orientate himself and others in the "pathless wilderness" of competing theories in electrodymanics around 1870; it was in this historical context that he promulgated his own contribution to the ongoing discussion about a fundamental potential for current elements. As already noted, those current potentials were mathematical tools used to derive further equations. Thus, the negative gradient of the potentials (the variation with repsect to changing position) furnished laws of ponderomotive forces, that is laws of mechanical forces between distant linear currents. The time derivative of the potentials furnished the electromotive force induced in systems of time-varint currents.

Page 11 of Eddington's Principle in the Philosophy of Science

In order to generate mechanical momentum, we usually need the action of a pondermotive force. Now a ponderomotive force of electromagnetic origin does act on conduction-current, but there is no conduc-tion-current in the free aether.

Page 165 of a 1922 Bulletin of the National Research Council By National Research Council (U.S.)

According to the Maxwell-Lorentz theory the fundamental equation for the calculation of all ponderomotive forces of electromagnetic origin is $f = q(E + \frac 1 c \vec v \times\vec H)$

So Minkowski meant the electromagnetic force on mass - the Lorentz force.
A: He just means the Lorentz force. The Lorentz force is called the "pondermotive force" in his paper, for no good reason. Old papers did not have internet to standardize their terminology for them.
A: Ron's responses are incorrect. You can see here that it was Boot and Harvie in 1957. In an inhomogenous plasma all particles regardless of charge will move toward the weaker field. This is much different from Lorentz forces, ie the motion of neutral particles do not generate a magnetic field.
