We always hear in classical mechanics that the interaction between particles happens instantaneously, and I think this assumption is obvious just by seeing Newton's third law. But I was wondering, is it possible to show that the velocity of this interaction is infinitely fast, just assuming the force is dependent only on the position of the particles (my professor said it is, but I can't see how to do it).
I couldn't find about it anywhere so, if anyone could help, I'd appreciate. Thank you.
When people say the force acting on particle 1 at some definite time depends only on positions of the other particles at the same time, this actually means there is no propagation of the interaction. The interaction is just everywhere and reflects current state of particles in the whole world. The concept of speed of interaction is superfluous.
"Infinite speed" of interaction is just a figure of speech that refers to this kind of theory.
By Newton's third law, action and reaction happens instantaneously if they are in contact with each other or at rest (in inertial frames). But, they do not hold true if you consider two moving objects, both traversing in orthogonal directions, for example, two charged particles moving in perpendicular directions away from each other. In such a case, it can be seen that a direct application of Newton's third law on the electromagnetic action and reaction on each particle violates the conservation of momentum of the system. This simply is due to the fact that nothing, nor even action or reaction can travel faster than light. So action and reaction interactions are not infinitely speedy. We have a universal speed limit.
Two charged particles interact via electromagnetic field, which happens at the speed of light (still it's finite, not infinite). But when the distance between the charges is sufficiently large, there is some delay, and we use the concepts of retarded potentials there.
Consider the force on a particle $B$ due to a particle $A$. The forces in newtonian physics, namely electromagnetism and gravity, have the following laws:
The acceleration of $B$ due to $A$ is
$$ \vec{a}_B = \frac{G m_A}{r_{AB}(t)^2} \hat{r}_{AB} + \frac{k q_B q_A}{m_B r_{AB}(t)^2} \hat{r}_{AB} $$
where $r_{AB}(t)$ is the distance between $A$ and $B$ at universal time slice $t$.
Therefore, the force on $B$ due to $A$ depends only on the state of $A$ at the "exact same time." You could imagine a force law that that propagated with finite speed by considering a force that depends on the position, or whatever other state variables, of $A$ at some time before $t$, $t'$, where $t' = t - \frac{r_{AB}}{c}$ for a force that propagates at velocity $c$.
Indeed, following this line of logic brings us to special relativity.
