## Interconnected Pendulums

**Problem**

The bobs of two pendulums are connected by a spring. When one is moved by an angle (theta) theta and released, what is the resulting motion? Do they eventually have the same energy?

For physics students, let the pendulums length=L, point mass=M, and spring constant=K. Find the equations of motion for both pendulums.

**Solution**

This is by far the most complicated problem that has ever been on problem of the week. Feel good if you were able to solve part of it. You likely wont be able to take the solution this far until college Physics. If you have a more intuitive solution, please add it to the comments.

For beginners, you’d do well to recognize that the energy will never be equally distributed between the two pendulums. It will be periodic, with one pendulum having all the energy at certain times and the other having none, and then switched.

For advanced students, the Lagrangian (kinetic energy minus potential energy) is given by:

where

The solution to the problem is then path which minimizes the Lagrangian, or the particular solution to the differential equation:

where q is a generalized coordinate. For anyone who would like an additional challenge, solve this equation and graph some results in MATLAB or your favorite mathematical language. Small angle assumptions may be used. This will be the Problem of the Week this week. Good luck.

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