Newton-Eulerian Axiomatic: The Notion of State

This chapter deals with the all-important notion of state as it is central to all parts of physics. For this book, most important are, on the one hand, the notions of state by Newton and Euler – on the other hand, the notion of state by Lagrange and Laplace used nowadays. Surprisingly enough, Newton and Euler’s notions of state are closer to quantum mechanics than the nowadays’ one. Newton’s axiomatic contains solely the conservation of states (1st and 3rd axioms) and the change of states (2nd axiom). In contrast, the equation of motion is not part of the axiomatic. Euler’s axiomatic contains solely the conservation of states. The change of state (and subsequently the equation of motion) is to be examined according to the problem under consideration. In contrast to Newton’s axiomatic and correcting Bohr’s corresponding claim, Euler’s principles of state change for classical bodies are formulated such, that they can be translated even to quantum-mechanical motion. Their power is also demonstrated through an alternative derivation of Hamilton’s equation of motion.

Keywords: d’Alembert’s principle, Axiomatic, Dynamics, Euler’s principles of stationary-state change, Hamiltonian, Inertia, Newton’s axioms, State, State function, Statics, Stationary state.