Non-inertial physics of action-angle variables of non-trivial periodic motion

Action-angle (AA) variables provide a framework to study periodic motion with the basic tools of Hamiltonian mechanics. They require a reference frame for their definition as physical observables (angles and angular momenta) in real space, but the nature of this frame is usually disregarded. We employ the active perspective of classical mechanics to demonstrate that the AA approach introduces a mapping between non-trivial periodic motion (i.e. when instantaneous frequencies deviate from the system’s characteristic frequencies) observed in an inertial frame and simple circular trajectories observed in a suitably chosen non-inertial frame where the AA variables can be defined as physical observables. After a general description of the critical aspects in the AA approach to non-trivial periodic motion, the link between AA variables and non-inertial frames is illustrated through the pedagogically rich example of the two-dimensional isotropic harmonic oscillator, a model representative of the broad class of Hamiltonians with non-trivial periodic signature. Our key conclusion is that the familiar representation of AA variables on tori in phase space is not merely a mathematical abstraction but corresponds to a topological transformation induced by the Euler force thanks to a change of reference frames in real space, from inertial to non-inertial. This work not only enhances students’ understanding of AA variables but also provides a pedagogical opportunity to underline the role of the non-inertial Hamiltonian mechanics, often limited to few examples free from the Euler force.