Effect of dissipation on the eigensolutions near the fission saddle point

Two equivalent methods are developed for solving the problem of small oscillations near equilibrium when dissipative forces are included in the dynamical equations of motion. One method relies on the Lagrangian formulation of the equations of motion, and the other method relies on the Hamiltonian formulation. The eigenvalues are in general complex, but for unstable or overdamped motion they are purely real. We use the Lagrangian formulation to calculate the two lowest symmetric eigensolutions at the fission saddle point for nuclei with $\frac{Z^2}{A}$ ranging from 18.0 to 44.6 for ordinary two-body viscosity, one-body wall-formula dissipation, and one-body wall-and-window dissipation.