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.
It's hard to think of a physical system involving a force that acted for zero time. However I think it's useful to consider a collision, perhaps between two billiard balls.
When the balls collide they change momentum. We know that the change of momentum is just the impulse, and we know that the impulse is given by:
$$ J = \int F(t)\,dt $$
where I've used an integral because the force is generally not be constant during the collision.
If we use soft squidgy balls then the collision will take a relatively long time as the balls touch, then compress each other, then separate again. If we use extremely hard balls the collision will take a much shorter time because the balls don't deform as much. With the soft balls we get a low force for a long time, with the hard balls we get a high force for a short time, but in both cases (assuming the collision is elastic) the impulse (and change of momentum) is the same.
When we (i.e. undergraduates) are calculating how the balls recoil we generally simplify the system and assume that the collision takes zero time. In this case we get the unphysical situation where the force is infinite but acts for zero time, but we don't care because we recognise it as the limiting case of increasing force for decreasing duraction and we know the impulse remains constant as we take this limit.
I'm not sure it's helpful to think about the gravitational force, because I can't see a similar physical system where we can imagine the gravitational force deliverting a non-zero impulse in zero time.
Response to edit:
In you edit you added:
If I got it right, you are saying that we must consider it impulse when t=0?, else it is force.
I am saying that if we use an idealised model where we take the limit of zero collision time the impulse remains a well defined quantity when the force does not.
However I must emphasise that this is an ideal never achieved in the real world. In the real collisions the force and impulse both remain well behaved functions of time and we can do our calculations using the force or using the impulse. We normally choose whichever is most convenient.
I think Mister Mystère offers another good example. If you're flying a spacecraft you might want to fire your rocket motor on a low setting for a long time or at maximum for a short time. In either case what you're normally trying to do is change your momentum, i.e. impulse, by a preset amount and it doesn't matter much how you fire your rockets as long as the impulse reaches the required value.
Response to response to edit:
I'm not sure I fully grasp what you mean regarding the book, but the force of gravity acting on the book does indeed produce an impulse. Suppose we drop the book and it falls for a time $t$. The force on the book is $mg$ so the impulse is:
$$ J = mgt $$
To see that this really is equal to the change in momentum we use the SUVAT equation:
$$ v = u + at $$
In this case we drop the book from rest so $u = 0$, and the acceleration $a$ is just the gravitational acceleration $g$, so after a time $t$ the velocity is:
$$ v = gt $$
Since the initial momentum was zero the change in momentum is $mv$ or:
$$ \Delta p = mgt $$
Which is exactly what we got when we calculated the impulse so $J = \Delta p$ as we expect.
Best Answer
In layman terms the Lorenz force is a first order equation in velocity. The charged particle has a constant v for the standard formula that gives the Lorenz force.
In nature and in the lab there exist nonlinear situations, where v is not a constant but changes. This has been studied and the second order solution for the motion of a particle in an electromagnetic potential A comes from a force which has been given the name of "ponderomotive force".
This definition is useful in laser plasma set ups, and for particle acceleration with lasers. See the link for details of derivation.
The mass enters because of the mathematics of the Maxwell's equations for the problem: charged particle in an inhomogeneous fields.