Which forces violate Newton's third law of motion?

Question

Forces arising from magnetic fields do violate Newton's third law of motion under certain circumstances. What other forces violate the third law?

Can you explain how does the magnetic force violate Newton's third law? — Harshit Joshi, Mar 19 '19 at 13:44
I would check this question out in dealing with magnetism and conservation of momentum, which is more fundamental than Newton's third law. — BioPhysicist, Mar 19 '19 at 13:45
@harshit54 The magnetic force between two moving particles can be found to not be "equal and opposite" — BioPhysicist, Mar 19 '19 at 13:46
There are only two forces which can be described in the classical approximation: electromagnetism and gravitation. As you have already realized, electromagnetic forces violate the third law of Newton. The gravitational forces do not violate the third law in the Newtonian approximation but they also violate the third law in general relativity. There are no other forces which can be seen at the classical level and the question of the third law of Newton become moot for them. — , Mar 19 '19 at 13:54
This appears to be a list-based question, which is generally considered off-topic as too broad. Please see the help center for details of what types of questions to avoid asking. — Kyle Kanos, Mar 19 '19 at 13:57
@KyleKanos Your point is correct but isn't the question still viable given that the list can maximally contain only $4$ items? — , Mar 19 '19 at 14:01
@DvijMankad IMO, no. It's simply asking for a list of things, rather than an explanation of things. — Kyle Kanos, Mar 19 '19 at 14:02
@KyleKanos I see. I had misunderstood the reasoning behind why one is not supposed to ask list-based questions. Thanks. — , Mar 19 '19 at 14:03
@AaronStevens The magnetic force doesn't act between two moving particles, though. It acts between each particle and the electromagnetic field. — probably_someone, Mar 19 '19 at 14:24
@DvijMankad your sticking at the "four fundamental interaction" does not fit very well with the variety of effective forces which are required and routinely used to model matter at different scales. Nobody could afford to solve the fundamental Hamiltonian of nuclei and electrons. Instead, adiabatic approximation, frozen core approximation, effective interactions like van der Waals, triple dipole, ionic models, three-body interactions are with us and provide good examples of interactions which may or may not obey Newton's third law. For further elaboration you may have a look ... — GiorgioP-DoomsdayClockIsAt-90, Mar 19 '19 at 20:25
@DvijMankad ... at my answer to another related question: https://physics.stackexchange.com/questions/443620/just-clearing-doubts-about-the-obviousness-of-newtons-laws — GiorgioP-DoomsdayClockIsAt-90, Mar 19 '19 at 20:26
@probably_someone Of course, I agree with your comment but I feel like adding that in a stationary set-up, it is fair enough to work with a model in which the magnetic forces are imagined to be acting between two moving charges--of course, the notion of "point finite charges" does not fair well with a stationary scenario but one can nonetheless work with a stationary set-up created by currents consisting of charged matter with finite charge density and treat an infinitesimal volume filled with such charge density as a point-like infinitesimal charge. — , Mar 19 '19 at 21:10
3rd law is violated in many cases, but a suitable reformulation (eg as conservation of momentum) still holds — Nikos M., Mar 03 '24 at 22:02

probably_someone · Accepted Answer · 2019-03-19T15:24:27.770

Anytime you hear that a force "violates Newton's Third Law," what that really means is that the action-reaction pair for that force has been misidentified. In the case of electromagnetic forces (for example,the case of two moving charges), it is incorrect to say that pairs of charges form action-reaction pairs. Rather, the action-reaction pairs are formed between each charge and the electromagnetic field itself. In electromagnetism, charges don't directly interact with other charges; rather, they interact with the electromagnetic field at their location. In other words, every action by the field on the charge $\mathbf{F}_{fq}$ has an equal and opposite reaction on the field by the charge $\mathbf{F}_{qf}$. Put formally, Newton's Third Law says that

$$\mathbf{F}_{fq}=-\mathbf{F}_{qf}$$

where

$$\mathbf{F}_{fq}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$$

and

$$\mathbf{F}_{qf}=\int \frac{d\mathbf{g}}{dt} dV=\int\frac{d}{dt}(\epsilon_0\mathbf{E}\times\mathbf{B})dV$$

where the integral is over all space. As you can see, the reason this works is because the electromagnetic field itself carries momentum density $\mathbf{g}$. Exerting a force on the field is equivalent to changing its total momentum.

Newton's Third Law is intimately related with conservation of momentum, and it is never violated as long as the interacting objects are correctly identified. If you incorrectly assume that two objects form an action-reaction pair (i.e. directly exert forces on each other), then Newton's Third Law may appear to be violated, but this is because the law has been misapplied.

Is the third law is a corollary of the conservation of angular momentum? I thought it is the other way around. In deriving the conservation of momentum of an isolated system of particles, you need to use the third law (for the cancellation of internal forces) to derive the conservation of momentum. @probably_someone — SRS, Mar 19 '19 at 14:59
@SRS It depends on how you're deriving classical mechanics. You could, for instance, start by assuming translational symmetry of the laws of physics, apply Noether's Theorem, and thereby obtain conservation of momentum, from which you can derive Newton's Third Law. Whether or not one is a corollary of the other depends on which set of initial assumptions you choose. — probably_someone, Mar 19 '19 at 15:16
@SRS I see your point, though. That sentence has been edited. — probably_someone, Mar 19 '19 at 15:24

score 3 · Answer 2 · answered Mar 19 '19 at 14:01

Any force which has a finite speed of propagation "violates Newton's Third Law" in this sense, at least momentarily. Imagine two particles $A$ and $B$ some large distance apart along the $x$-axis, exerting a central force on each other. Since there is a finite speed of propagation of the force, each particle feels a force according to the position of the other particle some time ago; if we imagine that these particles have been at rest for a long time, though, then their positions "some time ago" are the same as their current positions.

Now move particle $A$ a small distance off the $x$-axis. When $A$ moves off-axis it will feel a force from particle $B$, which is still on the $x$-axis; and since the force is central, this means that $A$ will feel a force that is not aligned in the $x$-direction. However, until the "news" of particle $A$'s displacement reaches $B$, $B$ will still feel a force pointing towards the former location of $A$, and this force will point along the $x$-axis. Since the forces on $A$ and $B$ do not point in the same direction, they can't possible be equal and opposite.

This discrepancy is usually addressed by taking into account the momentum of the field that mediates the force between $A$ and $B$. When we move $A$, it creates waves in this field, and these waves carry momentum. Assuming that the laws governing this field do not have any explicit dependence on position, it can be shown that the momentum of the particle plus the momentum of the waves obeys $\vec{F}_\text{tot} = d\vec{p}/dt$. Since Newton's Third Law is equivalent to saying that momentum is conserved in closed systems, it is rescued once we take the field momentum into account.

Which forces violate Newton's third law of motion?

2 Answers2