Question

mathematical derivation of newton's 3rd law of motion.????

Answer 1

Griffiths says "unlike the first two, Newton's third law does not, in general, extend to the relativistic domain." I don't understand, what is wrong with the following derivation? So we take it as an experimental fact that momentum is conserved in isolated systems (for clarity, the momentum we're working with is the spatial part of the 4-vector momentum or momenergy). Now imagine an isolated system consisting of two particles. Call ⃗ P P→ the momentum of the whole system, and ⃗ p 1 p→1 and ⃗ p 2 p→2 those of the respective particles in the system. By definition, ⃗ P = ⃗ p 1 + ⃗ p 2 P→=p→1+p→2 . Differentiate both sides with respect to coordinate time, then by definition of ordinary force: d ⃗ P d t = ⃗ F ext = ⃗ F 12 + ⃗ F 21 = d ⃗ p 1 d t + d ⃗ p 2 d t dP→dt=F→ext=F→12+F→21=dp→1dt+dp→2dt . Due to the experimental fact that ⃗ P = constant ⇒ ⃗ F ext = 0 P→=constant⇒F→ext=0 and thus as a result of the boxed equation: ⃗ F 12 = − ⃗ F 21 F→12=−F→21 .

Reference https://www.physicsforums.com/threads/derivation-of-newtons-third-law.499118/

Answer 2

Explanation:

Newton's third law states that :For every action there is equal and opposite reaction .

PROVE:

Let the object be F and the forces be a and b

so according to Newton's third law of motion

Fa = -Fb