The Adiabatic Invariance of the Action Variable in Classical Dynamics

Abstract

We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well-known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We present a new proof of the adiabatic invariance of this quantity and illustrate our arguments by means of explicit calculations for the harmonic oscillator. The new proof makes essential use of the Hamiltonian formalism. The key step is the introduction of a slowly-varying quantity closely related to the action variable. This new quantity arises naturally within the Hamiltonian framework as follows: a canonical transformation is first performed to convert the system to action-angle coordinates; then the new quantity is constructed as an action integral (effectively a new action variable) using the new coordinates. The integration required for this construction provides, in a natural way, the averaging procedure introduced in other proofs, though here it is an average in phase space rather than over time.

8 pages

Cited by in corpus (7)

• The Measure Problem in Cosmology
• On the identification of substructure in phase-space using orbital frequencies
• Pinning an Ion with an Intracavity Optical Lattice
• Periodicity and area spectrum of black holes
• Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants
• Variations on the adiabatic invariance: the Lorentz pendulum
• Action-angle variables for the purely nonlinear oscillator