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EQ of Motion
The Lagrangian of a dynamical system is a function that summarizes the dynamics of the system. The Lagrangian is named after Italian-French mathematician and astronomer Joseph Louis Lagrange.
Cartesian coordinates of M and m with respect to the pivot point of the first pendulum
Lagrangian
Velocities^2
Coordinates
Finding the Frequencies and
Normal Modes
Through some algebra and simplification the normal modes can be obtained
Putting the equations in matrix form and take the determinant to find the eigenvales - the frequencies
Convert the coupled differential equations into a system of linear algebraic equations by substitution
Special Cases
In both modes, the heavier upper mass essentially stands still while the lighter lower mass oscillates, and even flips around the upper pendulum.
Image by Michael Devereux
In the first mode (the positive one) the lower heavier mass is essentially standing still while the upper lighter mass vibrates back and forth at a high frequency due to the tension in the rods. In the second mode, the rods form a straight line and the system is essentially a simple pendulum with length 2 l
Long exposure of 2 LEDs, attached to the end of each pendulum of a double pendulum.
In the first mode the masses are moving equal distances in opposite directions at a high frequency. In the second mode the rods form a straight line, similarly to when M was less than m, and the system acts like a single pendulum.
In the first mode the bottom mass is basically standing still while the top mass is oscillating at a high frequency. In the second mode the system acts like a pendulum of length l because the tangential force on the top mass is zero.
A chaotic system is one that exhibits aperiodic long term behavior in a determinist system with sensitivity to initial conditions
Single Pendulum
Double Pendulum
Sources