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Proving Newton's third law invariant with Galilean tranfrom

  1. Jan 30, 2015 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    Newton's third law:
    F12 = -F21

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

    3. 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)
     
  2. jcsd
  3. Jan 30, 2015 #2
    Your left hand side has a second order differential equation of a sum. You can use the sum rule and work that out
     
  4. Jan 30, 2015 #3
    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?
     
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