>>12544964With
with r=r(t) and with the prime denoting time derivative, so I don't have to look up how to make a double dot.
You find
and imposing
(Newton 2 for potentials)
sets this zero.
As the derivative is linear, you can also play the game with a sum .
From then on it's a wish.
It's similarly enforced via the quantum mechanics axioms.
The analytical mechanics framework allows you for broader classes of models where the conserved quantity (the Hamiltonian) need not be T+V, but those models are usually very explicit about why that happens (where full Newtonian modeling is dropped - e.g. statics)
