ADIABATIC INVARIANCE DERIVED FROM THE RUND-TRAUTMAN IDENTITY AND NOETHER THEOREM

The generators of the infinitesimal transformations that lead to adiab atic invariance are derived from the Rund-Trautman identity by solving the Killing equations for a fairly generic class of models. Noether's theorem then yields the conserved quantity, which for periodic motion with period T is [H]T, where [H] is the time average of the Hamiltoni an over one cycle. Further it is shown that if [H]T is adiabatically i nvariant then so is the action closed-integral pdq, as the two differ by an invariant constant. Our approach (1) requires essentially no new concepts beyond those of a junior-level mechanics course, (2) shows h ow adiabatic invariance fits into the larger picture of the general co nnection between invariances and conservation laws, and (3) not only c onfirms and generalizes the results of Boltzmann, Clausius, and Ehrenf est for what is adiabatically invariant, but also predicts the rescali ng transformations that lead to this type of invariance.