Double Pendulum
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 20
28-7-2016
1256
Double Pendulum
The double pendulum consists of a mass m suspended by a massless string or rod of length l, from which is suspended another such rod and mass (see Figure 1.1).
Figure 1.1
a) Write the Lagrangian of the system for θ1, θ2 << 1.
b) Derive the equations of motion.
c) Find the eigenfrequencies.
SOLUTION
a) For the first mass m, the Lagrangian is given by
ignoring the constant mgl. To 
find introduce the coordinates for the second mass (see Figure 1.2):
Figure 1.2
Now, 
where
So
For θ1, θ2 << 1, we can take cos θ = 1- θ2/2. Denoting the frequency of a single pendulum by 
and eliminating superfluous constant terms, we obtain the Lagrangian in the form
(1)
b) Using (1) we can write the equations of motion
(2)
c) We are looking for solutions of (2) of the form
(3)
After substituting (3) into (2), we get a pair of linear equations in A and B
(4)
For nontrivial solutions of (4) to exist, we should have
(5)
The eigenfrequencies are defined from
(6)
Finally,
(7)
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