  The paper provides a new framework for the description of linearized
adiabatic lagrangian perturbations and stability of differentially rotating
newtonian stars. In doing so it overcomes problems in a previous framework by
Dyson and Schutz and provides the basis of a rigorous analysis of the stability
of such stars. For this the governing equation of the oscillations is written
as a first order system in time. From that system the generator of time
evolution is read off and a Hilbert space is given where it generates a
strongly continuous group. As a consequence the governing equation has a
well-posed initial value problem. The spectrum of the generator relevant for
stability considerations is shown to be equal to the spectrum of an operator
polynomial whose coefficients can be read off from the governing equation.
Finally, we give for the first time sufficient criteria for stability in the
form of inequalities for the coefficients of the polynomial. These show that a
negative canonical energy of the star does not necessarily indicate
instability. It is still unclear whether these criteria are strong enough to
prove stability for realistic stars.
