  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.
