Conservation of momentum implies Newton's third law of motion
the verdict
CONTESTED PARTIAL
refutedsupported
the weight of evidence
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Peer-reviewed literature indicates that conservation of momentum and Newton's third law are equivalent only under specific conditions rather than universally in all general systems.
The relation between momentum conservation and Newton's third law revisited
Under certain conditions usually fulfilled in classical mechanics, the principle of conservation of linear momentum and Newton's third law are equivalent. However, the demonstration of this fact is usually incomplete in textbooks. We shall show here that to demonstrate the equivalence, we require the explicit use of the principle of superposition contained in Newton's second law. On the other hand, under some additional conditions the combined laws of conservation of linear and angular momentum, are equivalent to Newton's third law with central forces. The conditions for such equivalence apply in many scenarios of classical mechanics; once again the principle of superposition contained in Newton's second law is the clue.
Published as: Revista Mexicana de Fisica E 51 N. 2 (2005) 99-101
arXiv categories: physics.class-ph physics.gen-ph
# Does conservation of momentum really imply Newton's third law?
Tags: newtonian-mechanics, education, momentum
- Score: 20
- Views: 8555
- Answers: 4
- Answered: yes
- Asked by: martin (1201 rep)
- Asked: 2011-10-25
- Edited: 2014-09-19
- Site: physics
## Question
I often heard that conservation of momentum is nothing else than Newton's third law.
Ok, If you have only two interacting particles in the universe, this seems to be quite obvious.
However if you have an isolated system of $n$ ($n > 2$) interacting particles (no external forces). Then clearly Newton's third law implies conservation of total momentum of the system. However presuppose conservation of total momentum you only get:
$$
\sum_{i\neq j}^n \mathbf F_{ij} = \frac{d}{d t} \mathbf P = 0
$$
Where $\mathbf F_{ij}$ is the forced acted by the $i$th particle upon the $j$th particle and $\mathbf P$ is the total linear momentum.
But this doesn't imply that $\mathbf F_{ij} = -\mathbf F_{ji}$ for $j \neq i$.
So does conservation of momentum implies Newton's third law in general or doesn't it? Why?
## Answers
### Answer by Luboš Motl (score: 7 [ACCEPTED])
Right, you could satisfy the momentum conservation by for