Conservation of Momentum and Lagrangian

[Ananthan9470]

In Leonard Susskind's the theoretical minimum, he says, "For any system of particles, if the Lagrangian is invariant under simultaneous translation of the positions of all particles, then momentum is conserved". For a system of two particles moving under a potential which is a function of the separation between the particles, he goes on proving that the total momentum will be conserved using symmetry. We know that the total momentum of a two particle system will be conserved if there is no external force acting on it from Newton. I am trying to understand how 'simultaneous translation of the positions of all particles' and 'potential is a function of the separation between the particles' is equivalent to saying there is no external force acting on the system. Any pointer will be helpful. Thanks!

 

[Orodruin]

You can rewrite the Lagrangian as a function of the CoM position and the separation. Naturally, there will be no potential term related to the CoM position and therefore the derivative of the potential wrt CoM position is zero, i.e., no force acting to change the motion of those coordinates.