Classical Mechanics from Energy Conservation or: Why not Momentum?

Abstract

It is demonstrated that energy conservation allows for a ``heuristic'' derivation of Newtonian mechanics, if the energy is presumed to be an additive function of position and velocity. It is shown that energy must be depicted as a function of position and momentum in order to allow for the correct relativistic equations. Accordingly it is argued that not only quantum theory but also special relativity is intrinsically Hamiltonian and requires a description by coordinates and momenta instead of coordinates and velocities. Furthermore it is argued that the usual historical order of the ``formulations'' of mechanics, from Newtonian via Lagrangian to Hamiltonian mechanics, is illogical and misleading. Therefore it should be reversed.