The conservation of the total energy of closed systems belongs to the most general laws of nature. This chapter treats first the common way to the mechanical one starting with Newton’s equation of motion. Then, the possibility to use the energy law as foundation of classical mechanics is explored, notably, Planck’s axioms, Euler’s second equation of stationary-state change, Carlson’s principle of the stationarity of total energy, and Leibniz’s theorem on the conservation of kinetic energy and its dual on the conservation of potential energy. The chapter concludes with considerations about the relationship between energy and extension in configuration space as known from the harmonic oscillator and the Kepler ellipses, and analogous considerations in momentum configuration space.
Keywords: Axiomatic, Configuration space, Energy conservation, Extension, Harmonic oscillator, Helmholtz, Hodograph, Kepler ellipse, Leibniz’s theorem, Momentum configuration space, Nemorarius’ theorem, Newton’s equation of motion, Schütz, State, State function, Stationary state.