The geometry of the classical action in phase space

Abstract

We present a geometric representation in phase space of the classical action. Specifically, we find three seemingly unrelated interpretations for the action as the sum of signed areas of shapes in phase space. By using the Poincaré–Cartan integral invariants in extended phase space, we are able to show the equivalence between all three. As a concrete example, we consider the 1D harmonic oscillator, but the results are general for arbitrary potentials and numbers of dimensions.