Lesson - Eigenvalues

Consider the following problem, which occurs frequently in both mathematics and physics:

Problems of this form are referred to as eigenvalue problems. Eigen is the German word for "specific", so literally translated eigenvalue means "specific value".

Usually, A in these problems is a matrix, however, it can also be an operator (a function that acts on a function, e.g. a partial derivative with respect to time). The way to read this problem is "For what values of the vector x will the operator A applied to x return a multiple of x, and what are those multiples?" The values of the vector x are the eigenvectors of the problem, and the multiples are the eigenvalues. They are the specific vectors and values that make the statement true.

Exercise

Consider the following problem: An artist is commissioned to design a public sculpture with a theme of pendulum motion. It is decided that the sculpture will feature four pendulums of different length swinging in motion, such that the longest pendulum takes 10 seconds to complete a swing, and that the other pendulums will in that time take 2, 3, and 4 swings respectively, but all will return to their original position every 10 seconds.

1. What are the eigenvalues that will satisfy the artist's design?
2. Before constructing the sculpture, the artist becomes concerned that the small angle approximation might not be valid in this case. How would relaxing the small angle approximation change your solution?

Use the numerical model of pendulum motion to solve this problem.