[ars]

Am I correct in understanding that this takes every kind of interaction that particles can have, and simply adds them all together?

How does it deal with some particles only interacting in certain ways and not others? Is the user of the equation supposed to make sure to enter zero for those terms, or does the equation capture that knowledge as well?

[sprash]

If you apply an operator of such an interaction to the Lagrangian the equations of motions automatically return zero (e.g. similar to a derivation of a constant).