See my recent papers on rotational dynamics of a rigid body around central but non-principal axis by dry friction acting. Due to d'Alembert principal there will be normal to the axis  inertia forces and hence the frictional axial  torques definig the angular acceleration and also depending on it. The reason and consequance are coupled in such way.

There is in fact much more than the parallelogram principle at stake here: if forces can actually depend on accelerations, then Newton's second law does not uniquely determine acceleration in terms of lower order derivatives, since it becomes a generally non-linear relation between acceleration, velocity and position. Multiple values of aceleration could then arise as a consequence for the same values of velocity and position, thus contradicting determinism. It might even happen that, even though initial conditions lead to unique acceleration as a function of velocity and position, it might suddenly happen that the equation turns into one that has no solution. Consider ddot x = F(x) + epsilon (ddot x)^2. If F(x) aloows rapid growth of x and if epsilon is small but constant, a catastrophe happens if F(x) allows for fast growth of x, as would happen with F(x)=lambda x, for lambda>0.

In fact, dry friction in its usual form is a force that depends on other forces (though not directly on acceleration) and this is enough to cause similar difficulties, as observed by Painlevé (so-called Painlevé paradoxes of dry friction).

And what about exept of dry friction? Сan such efforts exist?

There are two questions here: do such forces exist experimentally? and are such force laws mathematically consistent? If the answer to the secon question is negative, then in fact the answer to the first cannot really be positive: indeed, if a mathematical reasoning leads to an absurdity, this cannot be what we observe. Now depending very much on the details of the way your force depends on the acceleration, you risk very much being led into contradictions. So if you ever wanted to propound a model with such forces, you should argue why it does not lead to paradoxes like the orbit suddenly stopping to be defined.

In the Painlevé paradoxes, it turns out that extremely high normal forces are generated in the paradoxical situations, so we may hope that the ordinary Coulomb law maintains its validity as long as normal forces are not too large.

Finally, to the best of my knowledge, the fundamental forces of nature never involve accelerations, and the approximate descriptions are usually framed in terms of equations with memory (which can then often be approximated in rurn by equations with forces dependent on position and velocity) but never involving accelerations.

Awesome question. I know different people have different views on how to interpret Newton's three laws. My favourite is that the definitions of time, length and the first law define acceleration, that mass is defined using the third law and comparing the accelerations of interacting particles, one of which is a reference (that mass is constant is just an experimental observation), and finally that force is *defined* as mass times acceleration F:=ma (J. B. Griffiths. The theory of classical mechanics. Cambridge University Press, 1985). Then, although convenient, maybe it was a mistake to introduce the concept of force?

With this view, force becomes redundant as we can always write ma instead; same thing. Particularly, one takes the view that F:=ma(t,r,v). We cannot have a(a) as it would redefine the concept of mass. The laws of nature could just as well be written in terms of acceleration instead of force, which is just a dummy quantity, and this is presumably the reason why the fundamental forces of nature never include acceleration.

(Newton, I think, at the time saw mass and force as predefined physical quantities that needed no further explanation/definition, but who knows what his position might be today).

Of course it is mistake to introduce force through acctltration. The force is a classical measure, extent of body interaction. All force may be ordered by magnitudes and direction in cardinal way, mainly through the deformations they induce. The mass is simply a number of all protons+ neutrons +1.1700 of that of electons. We don't need any acceleration to introduce such an obvious value as a quality of the matter. Very supprised by quotation of Griffiths et al.

Viktor: I have made an attempt to understand your paper (with Google translate, because I don't know Russian!) I think I have understood the basic idea, and I do agree with you: in this case, due to dry friction, we in fact have a nontrivial equation connecting the acceleration with itself. But this is a well-known peculiarity of dry friction, in which the force is expressed in term of other forces. And in fact the paradoxes which I alluded to above are well known in the case of dry friction and are known as the Painlevé paradoxes.

As for your definition of mass, I tend to be on your side. But the issue is a bit more complex: how do you know that the mass of an electron is 1800 times less than a proton? You need a definition of mass. So you say: the same force generates 1800 times less acceleration in a proton. OK, but what is a force, and how do you say that two forces are the same? If you define force as mass*acceleration you hve a circular definition, A rather good way out is to do things using the third law: if a proton collides with an electron we say *by definition* that the force the proton exerts on the electron is the same as viceversa and then observe experimentally that the resulting acceleration is 1800 times less in the proton than in the electron.

Dear F,!

1. No need to define mass through acceleration - it's better to define it as a gravitational one. To my mind the mass may be introduced as a quantity of matter in way like Kolmogorov probability. We order all material objects by this characteristic and this relation of equivavlence divides all the objects into classes of having equal masses. Then we introduce  first scale of ratios (logarithmic) then cardinal scale for m. No experiments with collisions etc are needed.( Better in magnet mass-spectrometer measuring the normal a) 

2 I asked about mechanical forces. As for non-mech. - electromagnetic ones. The charged particle irradiates with the intensity proport, a^2 therefore the delaying force is prop. a^2..