a double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. if the two pendula have equal lengths, $l$, and have bobs of equal mass, $m$, and if both pendula are confined to move in the same vertical plane, find lagrange's equations of motion for the system. use $\theta $ and $\phi$--the angles the upper and lower pendulums make with the downward vertical (respectively)--as the generalized coordinates. do not assume small angles.

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09-12-2022
Physics

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a double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. if the two pendula have equal lengths, $l$, and have bobs of equal mass, $m$, and if both pendula are confined to move in the same vertical plane, find lagrange's equations of motion for the system. use $\theta $ and $\phi$--the angles the upper and lower pendulums make with the downward vertical (respectively)--as the generalized coordinates. do not assume small angles.

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kjuli1766 kjuli1766 · Answer 1 · 2022-12-18T04:14:36-05:00

The Lagrange's equations of motion for the system is d(ml³Ф2 + ml²Ф1 cos (Ф1-Ф2)) − (−ml³Ф1Ф2 sin (Ф1-Ф2)-mgl sinФ2 )/dt = 0.

Newton's method of developing the equations of motion requires element decomposition. If the forces on the connections are not the primary concern, it is more advantageous to consider the energies in the system to derive the equations of motion.

A double pendulum exhibits simple harmonic motion when the non-equilibrium displacement is small. However, when large displacements are imposed, the behavior of nonlinear systems becomes dramatically chaotic indicating that deterministic systems are not always predictable. There are several possible variations of the double pendulum.

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