Mathematics Math21b Spring 2008
Linear Algebra and Differential Equations
Exhibit: double pendulum
Office: SciCtr 434
The pendulum motion is described by the differential equation
x''(t) = -sin(x)
where x(t) is the angle at time t. For small x, the linearization
x''(t) = -x
is a good approximation.
The picture shows the double pendulum borrowed from the Harvard
The coupled system of two pendula is
x''(t) = - x + c (y-x)
y''(t) = - y - c (y-x)
The constant c is determined by the spring strength.
Solving the system
This system in vector form is
v''(t) = A v(t)
with the matrix
A = | -(1+c) c |
| c -(1+c) |
This matrix has the eigenvalues L1 = -1 with eigenvector v2 = [1,1]T and
the eigenvalue L2 = -1-2c 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) = a1 v1 + a2 v2
v'(0) = b1 v1 + b2 v2
This produces the solution
v(t) = [ a1 cos(t) + b1 sin(t) ] v1
+ [ a2 cos((1+2c)1/2 t) + b2 sin((1+2c)1/2 t) ] v2
In 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 bi=0 and a2=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 bi=0 and a1=0, the pendulum swing against each other.
The frequency (1+2c)1/2 has become slightly larger.