The dots above q1 and q2 represent the time derivative of q1 and q2, respectively.  The addition of negative lambdas (or Eigenvalues) to the main determinants enable us to solve for these values and consequently for q1 and q2. The values of q1 and q2 can be obtain from the following equations:

C1 and C2 are determined from initial conditions.   For example, the mass in C1 and C2 at initial time.