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This question is never addressed by teachers, altough students start asking it more and more every year (surprisingly). Here are two possible arguments.

1. A particle is meant to have 0 volume. Maybe you're used to exert a force on yourself, but you are an extended body. Particles are points in space. I find it quite hard to exert a force on the same point. Your stating that the sender is the same as the receiver. It's like saying that one point is gaining momentum from itself! Because forces are a gain in momentum, after all. So how can we expect that some point increases its momentum alone? That violates the conservation of momentum principle.



2. 
   

   A visual example (because this question usually arises in Electromagnetism with Coulomb's law):

   
   
   

   $$vec{F}=K frac{Qq}{r^2} hat{r}$$

   
   

If $r=0$, the force is not defined, what's more, the vector $hat{r}$ doesn't even exist. How could such force "know" where to point to? A point is spherically symmetric. What "arrow" (vector) would the force follow? If all directions are equivalent...

Well a point particle is just an idealization that has spherical symmetry, and we can imagine that in reality we have some finite volume associated with the "point", in which the total charge is distributed. The argument, at least in electromagnetism, is that the spherical symmetry of the charge together with its own spherically symmetric field will lead to a cancellation when computing the total force of the field on the charge distribution.

So we relax the idealization of a point particle and think of it as a little ball with radius $a$ and some uniform charge distribution: $rho= rho_{o}$ for $r<{a}$, and $rho=0$ otherwise.

We first consider the $r<a$ region and draw a nice little Gaussian sphere of radius $r$ inside of the ball. We have: $$int_{} vec{E}cdot{dvec{A}} =dfrac{Q_{enc}}{epsilon_{0}}$$ $$4pi r^{2}E(r) = frac{1}{epsilon_{0}}frac{4}{3}pi r^{3}rho_{0}
qquad , qquad r<a$$

Now we say that the total charge in this ball is $q=frac{4}{3}pi r^{3}rho_{0}$, then we can take the previous line and do $$4pi r^{2}E(r) = frac{1}{epsilon_{0}}frac{4}{3}pi a^{3}*frac{r^{3}}{a^3}rho_{0}=frac{q}{epsilon_0}frac{r^{3}}{a^{3}}rho_0$$

or

$$vec{E}(r)=frac{q}{4piepsilon_{0}}frac{r}{a^{3}}hat{r} qquad,qquad r<a$$

Outside the ball, we have the usual:
$$vec{E}(r)=frac{q}{4piepsilon_{0}}frac{1}{r^{2}}hat{r} qquad,qquad r>a$$

So we see that even if the ball has a finite volume, it still looks like a point generating a spherically symmetric field if we're looking from the outside. This justifies our treatment of a point charge as a spherical distribution of charge instead (the point limit is just when $a$ goes to $0$).

Now we've established that the field that this finite-sized ball generates is also spherically symmetric, with the origin taken to be the origin of the ball. Since we now have a spherically symmetric charge distribution, centered at the origin of a spherically symmetric field, then the force that charge distribution feels from its own field is now

$$vec{F}=int vec{E} , dq =int_{sphere}vec{E} rho dV = int_{sphere} E(r)hat{r}rho dV$$

which will cancel due to spherical symmetry. I think this argument works in most cases where we have a spherically symmetric interaction (Coulomb, gravitational, etc).

What even is a particle in classical mechanics?

Particles do exist in the real world, but their discovery pretty much made the invention of quantum mechanics necessary.

So to answer this question, you have to set up some straw man of a "classical mechanics particle" and then destroy that. For instance, we may pretend that atoms have the exact same properties as the bulk material, they're just for inexplicable reasons indivisible.

At this point, we cannot say any more whether particles do or do not exert forces on themselves. The particle might exert a gravitational force on itself, compressing it every so slightly. We could not detect this force, because it would always be there and it would linearly add up with other forces. Instead, this force would show up as part of the physical properties of the material, in particular its density. And in classical mechanics, those properties are mostly treated as constants of nature.

This exact question is considered at the end of Jackson's (somewhat infamous) Classical Electrodynamics. I think it would be appropriate to simply quote the relevant passage:

In the preceding chapters the problems of electrodynamics have been
divided into two classes: one in which the sources of charge and
current are specified and the resulting electromagnetic fields are
calculated, and the other in which the external electromagnetic fields
are specified and the motions of charged particles or currents are
calculated...

It is evident that this manner of handling problems in
electrodynamics can be of only approximate validity. The motion of
charged particles in external force fields necessarily involves the
emission of radiation whenever the charges are accelerated. The
emitted radiation carries off energy, momentum, and angular momentum
and so must influence the subsequent motion of the charged particles.
Consequently the motion of the sources of radiation is determined, in
part, by the manner of emission of the radiation. A correct treatment
must include the reaction of the radiation on the motion of the
sources.

Why is it that we have taken so long in our discussion of
electrodynamics to face this fact? Why is it that many answers
calculated in an apparently erroneous way agree so well with
experiment? A partial answer to the first question lies in the second.
There are very many problems in electrodynamics that can be put with
negligible error into one of the two categories described in the first
paragraph. Hence it is worthwhile discussing them without the
added and unnecessary complication of including reaction effects. The
remaining answer to the first question is that a completely
satisfactory classical treatment of the reactive effects of radiation
does not exist. The difficulties presented by this problem touch one
of the most fundamental aspects of physics, the nature of an
elementary particle. Although partial solutions, workable within
limited areas, can be given, the basic problem remains unsolved.

There are ways to try to handle these self-interactions in the classical context which he discusses in this chapter, i.e. the Abraham-Lorentz force, but it is not fully satisfactory.

However, a naive answer to the question is that really particles are excitations of fields, classical mechanics is simply a certain limit of quantum field theory, and therefore these self-interactions should be considered within that context. This is also not entirely satisfactory, as in quantum field theory it is assumed that the fields interact with themselves, and this interaction is treated only perturbatively. Ultimately there is no universally-accepted, non-perturbative description of what these interactions really are, though string theorists might disagree with me there.

This answer may a bit technical but the clearest argument that there is always self interaction, that is, a force of a particle on itself comes from lagrangian formalism. If we calculate the EM potential of a charge then the source of the potential, the charge, is given by $q=dL/dV$. This means that $L$ must contain a self interaction term $qV$, which leads to a self force. This is true in classical and in quantum electrodynamics. If this term were absent the charge would have no field at all!

In classical ED the self force is ignored, because attempts to describe have so far been problematic. In QED it gives rise to infinities. Renormalisation techniques in QED are successfully used to tame the infinities and extract physically meaningful, even very accurate effects so called radiation effects originating from the self interaction.

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.

John David Jackson, Classical Electrodynamics.