Pendulum without friction. Limit cycle in phase space: no sensitivity to initial conditions. Pendulum with friction. Fixed point attractor in phase space: no sensitivity to initial conditions. Pendulum with friction: basin of attraction.

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Double Pendulum. Double Pendulum. The double pendulum is a conservative system. Two degrees of freedom The exact Lagrangian can be written without approximation. l. q. m. l. f. m. Make substitutions: Divide by mgl t t ( g / l ) 1/2. Find conjugate momenta as angular momenta.

Double Pendulum. By: Rosa Nguyen EPS 109 – Fall 2011. Double Pendulum. - A simple pendulum where another point of movement can now occur (or another rod added at the end).

Double Pendulum. Coupled Motion. Two plane pendulums of the same mass and length. Coupled potentials The displacement of one influences the other Coupling is small Define two angles q 1 , q 2 as generalized variables. k. l. l. q 1. q 2. m. m. Coupled Equations.

The Double Pendulum. by Franziska von Herrath & Scott Mandell. The potential energy of the system is found by combining the potential energy of each mass. The kinetic energy of the system is found by combining the kinetic energy of each mass. The Lagrangian of the system is:.

The Double Pendulum. by Franziska von Herrath & Scott Mandell. The potential energy of the system is found by combining the potential energy of each mass. The kinetic energy of the system is found by combining the kinetic energy of each mass. The Lagrangian of the system is:.

Some analytical models of nonlinear physical systems. P. 2. l. x. A. 1. k 2. l. g. k 1. O. y. Discrete Dynamical Systems : Inverted Double Pendulum : Consider the double pendulum shown:. Inverted Double Pendulum – E quation of motion.