# Proving Newton's third law invariant with Galilean tranfrom

• AllRelative
In summary, by using the sum rule for second order differential equations, it can be shown that Newtonian mechanics is form invariant with respect to a Galilean transformation. This means that the force law and equations of motion remain the same, regardless of the reference frame.

## Homework Statement

Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21.

m1(d^2r1/dt^2) = F12(r1,r2,u1,u2)

m2(d^2r2/dt^2) = F21(r1,r2,u1,u2)
A. Show that Newtonian mechanics is form invariant with respect to a Galilean transformation?

## Homework Equations

Newton's third law:
F12 = -F21

Galilean Transform:
t'=t
m'=m
r'=r-vt

## The Attempt at a Solution

I understand the concept of invariance. I understand Galilean relativity quite well. I just have no clue where to start the proof. I haven't done much proofs in my past physics and math courses.

I tried writing the two F=m*a equations with the primed coordinates but after that I'm lost. here is what I wrote:

m1(d^2(r1-v*t)/dt^2) = (F12)'((r1-v*t),(r2-v*t),(u1-v),(u2-v)

m2(d^2(r2-v*t)/dt^2) = -(F21)'((r1-v*t),(r2-v*t),(u1-v),(u2-v)

Your left hand side has a second order differential equation of a sum. You can use the sum rule and work that out

Thanks for the quick reply! Doing that brings me to:

m1(d^2(r1)/dt^2) - m1(d^2(v*t)/dt^2) =

m1*a1 - 0 = (F12)'((r1-v*t),(r2-v*t),(u1-v),(u2-v)

And therefore I can say that F12(r1,r2,u1,u2) = (F12)'((r1-v*t),(r2-v*t),(u1-v),(u2-v)

Is that right?