Thank you jtbell, that's very helpful. I myself am actually in the process of programming a simulator for the evolution of particles under gravity. That is actually where I encountered this problem. I noticed that when particles started to fall towards each other they would pick up speed and...
The lagrangian is different for different systems, the logic here is simply that if take it as a given that space is homogenous then it equivalent to say that the lagrangian cannot depend on the raw position. I think the problem you are having is that you are trying to go from the action...
Take a closed system of particles. Each has it's own mass, but that's really about it for the sakes of this argument. Now suppose there is some standard force law that applies between particles, whatever it may be. (In general it will depend only on the separation between particles and thus be...
Take a hypothetical object with an infinite density (purely so that a particle can get incredibly close to it). Furthermore imagine that this object has no close range repulsive force, or in other words, particles can pass right through it. The only thing is does is exert a standard...