# Galilei Invariance and Newton Third Law

Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized potential.

Assuming Isotropy of space and homogeneity of space and time, what are the constraints imposed on the possible forces between the particles? In particular, can Newton's third law be "derived" under such conditions?

• Newton's third law is conservation of momentum, which is equivalent to translational invariance by Noether's theorem -- so yes, this follows from your homogeneity assumptions. – Abhimanyu Pallavi Sudhir Mar 30 '19 at 5:07
• Indeed. But, for example, in the case of electromagnetic interaction there is no Newton third law and you need to consider the momentum of the field to have conservation of momentum. In the most general case of position and velocity dependent forces, what can be said about the forces if we assume homogenity and isotropy of space? – AndresB Mar 30 '19 at 13:29
• An important point to note would be that isotropy of space and homogeneity of space and time (and the invariance of physics among all inertial frames) do not ensure Galilean invariance. It very much leaves out space for Lorentz invariance. And, it is more than clear that under Lorentz invariance, any significant content of the third law cannot be correct due to the finite speed limit (or, say, due to the frame-dependence of simultaneity). Of course, even Galilean invariance doesn't ensure the third law as @ElioFrabri explains in their answer but Lorentz invariance explicitly rules it out. – Dvij D.C. Mar 31 '19 at 13:53

$$\def\br{{\bf r}} \def\bF{{\bf F}}$$ Consider the following system. Two particles, equal masses, no external forces. Force acting on particle #1 (due to particle #2): $$\bF_1 = k_1 (\br_2 - \br_1).$$ Force acting on particle #2 (due to particle #1): $$\bF_2 = k_2 (\br_1 - \br_2).$$

You can verify that this system satisfies

• isotropy of space (each force is always directed towards the other particle)
• homogeneity of space (a translation of system leaves forces invariant)
• homogemeity of time (forces do no depend on time).

Yet Newton's third law isn't satisfied if $$k_1\ne k_2$$. Total momentum isn't conserved, com is accelerated...

How can it be? The point is that @AbhimanyuPallaviSudhir is wrong: conservation of momentum is not equivalent to translational invariance. Or, to be more precise: it's not equivalent to translation invariance of forces - invariance of Lagrangian is required. Only if there is an invariant Lagrangian Noether's theorem can be proven.

But the system I defined admits of no Lagrangian. Actually its forces don't derive from a potential.

• +1. To be fair tho, I don't think @AbhimanyuPallaviSudhir explicitly said that the conservation of momentum is implied by the translational invariance of forces. What they got wrong for sure is that they seemed to think that the third law should follow from the conservation of momentum. Even in theories that admit a Lagrangian description and have a translational invariance, Newton's third law is violated left and right, to wit, electromagnetic theory. Relevant link: physics.stackexchange.com/questions/114466/… – Dvij D.C. Mar 31 '19 at 13:47
• @DvijMankad I don't think AbhimanyuPallaviSudhir explicitly said that the conservation of momentum is implied by the translational invariance of forces. Well, he wrote Newton's third law is conservation of momentum, which is equivalent to translational invariance by Noether's theorem. My aim was to stress that momentum conservation doesn't follow from translational invariance unless you don't make explicit what is too often left understood: that a Lagrangian is required. – Elio Fabri Mar 31 '19 at 13:57