References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.2, problem 1.
The nature of the dependence of a force or potential on the underlying position coordinates can determine certain conservation laws. In his chapter I.2, Zee shows that a central force (a force that is always directed towards its source, such as the Earth’s gravity or a point charge’s electrostatic field) conserves angular momentum. Actually his derivation is a generalization to any number dimensions of the more familiar proof in 3-d, which goes like this:
The angular momentum of a mass is defined as
where is the linear momentum. Taking the time derivative we get
where the third line uses the fact that for a central force, so their cross product is zero. Thus doesn’t change with time.
Now suppose that a force is the negative gradient of a potential function so that by Newton’s law:
where the index refers to particle in a collection of interacting particles, and is the component of the coordinate . Note that the in represents all components of (if we’re doing the calculation in -dimensional space) and not just the magnitude of the distance.
Now suppose that is a function only of the coordinate differences between particles and , where with . In this case, the total linear momentum is given by
The time derivative is
Since is a function of the set of all possible differences , the terms in the sum 9 cancel in pairs. For example, for 3 particles if , then will contain a term equivalent to (plus another term resulting from ). However, will contain a term equivalent to which cancels the first term. That is
so adding up all the derivatives causes the terms to cancel in pairs. The argument is fairly easily extended to particles.
Thus and linear momentum is conserved. [I realize this isn’t a very elegant or mathematical way of proving it; there is probably a better way of writing it down, but hopefully you get the idea.]