M  A  T  H 
2  1  B 

PendulumThe pendulum motion is described by the differential equationx''(t) = sin(x)where x(t) is the angle at time t. For small x, the linearization x''(t) = xis a good approximation. Double pendulumThe picture shows the double pendulum borrowed from the Harvard lecture demonstrations. The coupled system of two pendula isx''(t) =  x + c (yx) y''(t) =  y  c (yx)The constant c is determined by the spring strength. 
Solving the systemThis system in vector form isv''(t) = A v(t)with the matrix A =  (1+c) c   c (1+c) This matrix has the eigenvalues L_{1} = 1 with eigenvector v_{2} = [1,1]^{T} and the eigenvalue L_{2} = 12c with the eigenvector [1,1]^{T}. Write the initial position v(0) = [x(0),y(0)]^{T} and initial velocity v'(0) = [x'(0),y'(0)]^{T} as linear combinations of eigenvectors. v(0) = a_{1} v_{1} + a_{2} v_{2} v'(0) = b_{1} v_{1} + b_{2} v_{2}This produces the solution v(t) = [ a_{1} cos(t) + b_{1} sin(t) ] v_{1} + [ a_{2} cos((1+2c)^{1/2} t) + b_{2} sin((1+2c)^{1/2} t) ] v_{2} 
EigenmodesIn this system we can "see" the eigenvalues as the squared frequencies of the eigenmodes. One can also see the eigenmodes.1) Starting with an initial condition where b_{i}=0 and a_{2}=0, we have a double pendulum where both pendula swing parallel. The spring strenght does not matter and the penduli move as if they were not connected. 2) Starting with an initial condition where b_{i}=0 and a_{1}=0, the pendulum swing against each other. The frequency (1+2c)^{1/2} has become slightly larger. 
The first eigenmode.  The second eigenmode. It swings a bit faster. The small spring strength makes it swing only slightly faster.  A general initial condition. 