Need an equation for this force of colliding objects question?

Question

Why is the sum of the forces on colliding objects zero? (momentum is conserved)

Answer 1

This follows from the homogeneity of space, a symmetry principle saying that no point in space is special, or that any point can be selected as the coordinate origin without changing the rules of physics.

In classical physics the forces between objects can be derived from a potential function depending on the spatial positions of the objects V(x1,x2,...,xn).

The force associated with position x_i is defined as F_i = -Grad_i V. Because the coordinate origin can be taken as any point in space, adding any vector constant u to all coordinates should have no effect on the value of the potential energy:

V(x1+u,x2+u...,x_n+u) = V(x1,...,x_n)
-->
Grad_u V(x1+u,x2+u...,x_n+u) = 0
-->
Grad_x1 V + ... + Grad_xn V = 0
-->
-[F_1 + F_2 +... + F_n] = 0

i.e. all forces add up to zero. The question as stated only applies to nonrelativistic physics (more generally the momentum of the force field itself also has to be taken into account).