### 3. WAVES AND MODES IN DISKS

The best way to obtain insight into the self-consistent excitations of a
disk is to replace the disk by a collection of spinning, concentric rings
with the same overall surface density and angular momentum
distribution. ^{(1)}
Each ring *i* is characterized by two coordinates, the polar angles
_{i} and _{i} associated with its
normal. For any set of values of
these angles and their time derivatives it is straightforward to determine
the system's gravitational potential energy *V*({_{i}, _{i}})
and it's kinetic energy *K*({_{i}, _{i}}), and thus to
determine the Lagrangian and equations of motion.

Now there are two ways forward. Either one finds the system's normal modes
by linearizing the equations of motion and assuming that all coordinates
vary harmonically
(Sparke 1984a).
Or one can integrate the equations of
motion from arbitrarily chosen initial conditions.

^{1}
May & James (1984)
have shown that such ring models
reproduce the dynamics of full *n*-body simulations over long
periods.