Spherical Pendulum

The top of a pendulum of length hangs from the origin. The mass at the bottom end of the pendulum has coordinates , , , where the vector from the origin to is at an angle θ to the negative axis. The spherical coordinates of are (, , ) with . The Lagrange function and equations give , , and . The integration constants are , , , and the angular momentum . The movement of the spherical pendulum is constrained to the spherical shell between and for all values. The pendulum cannot reach the singular points and for . When the angular momentum vanishes, the pendulum moves in a plane.

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