  We study the nature of non-axisymmetric dynamical instabilities in
differentially rotating stars with both linear eigenmode analysis and
hydrodynamic simulations in Newtonian gravity. We especially investigate the
following three types of instability; the one-armed spiral instability, the low
T/|W| bar instability, and the high T/|W| bar instability, where T is the
rotational kinetic energy and W is the gravitational potential energy. The
nature of the dynamical instabilities is clarified by using a canonical angular
momentum as a diagnostic. We find that the one-armed spiral and the low T/|W|
bar instabilities occur around the corotation radius, and they grow through the
inflow of canonical angular momentum around the corotation radius. The result
is a clear contrast to that of a classical dynamical bar instability in high
T/|W|. We also discuss the feature of gravitational waves generated from these
three types of instability.
