Classical Mechanics from Energy Conservation or: Why not Momentum?

TL;DR

This paper shows that classical mechanics can be derived from energy conservation alone and emphasizes the fundamental role of Hamiltonian formalism in both quantum and relativistic theories, challenging traditional historical order.

Contribution

It proposes deriving Newtonian mechanics from energy conservation without work, and argues for reversing the historical development of mechanics formulations.

Findings

Energy conservation suffices for Newtonian mechanics derivation.

Energy must be a function of position and momentum for relativistic equations.

Quantum and relativistic theories are inherently Hamiltonian.

Abstract

It is demonstrated that energy conservation allows for a straight derivation of Newtonian mechanics without an apriori definition of the concept of work. Furthermore it is shown that energy must be depicted as a function of position and momentum in order to obtain the correct relativistic equations. Accordingly it is argued that not only quantum theory but also special relativity is intrinsically a Hamiltonian theory which requires a description of the dynamics using coordinate and momentum instead of velocity. 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. We suggest that it should be reversed.