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?
$\def\br{{\bf r}} \def\bF{{\bf F}}$ 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
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.
