Newton tried to do it with three laws/statements. While the first can be derived
ID: 2283636 • Letter: N
Question
Newton tried to do it with three laws/statements. While the first can be derived from the second, the three form a pretty nice framework.
Later on, I've encountered Lagrangian Mechanics, which involves, from what I gather, one statement:
Objects seek a path that minimizes total action.
Which, while sort of simple, doesn't sound too natural or fundamental.
I've heard some of these rather elegant statements:
The universe is translation-invariant
The universe is invariant under different inertial frames of reference
And from either of these, one may derive Newton's laws and F=ma. However, I'm not completely sure how this can be done. Were these claims true? How is this possible?
Has anyone ever come across a surprisingly/particularly stunningly elegant/fundamental formulation of classical mechanics, and a proof that they are equivalent to Newtonian Mechanics?
Explanation / Answer
The principle of least action is much more elegant and fundamental than the explicit formulation of the laws of mechanics using three laws of Newton which disproves your intuition that the principle of least action is not elegant or fundamental.
The two symmetries you mention are symmetries that constrain the evolution of physical systems but they don't totally determine what the evolution looks like. It is not true that the symmetries you mention are enough to derive the forces among a set of objects. F=ma, without specifying how F depends on the positions (e.g. gravitational force), is just a definition of F and it is a vacuous statement so you may derive it from anything - even from no assumptions at all.