370(6): General Transformation from Spherical Polar to Eulerian Angles

This is given by Eq. (4) in general, and in the special case (5), by Eq. (6). It is difficult to find this transformation in the literature, but it is very useful. The lagrangian of rotational dynamics in terms of the Euler angles can always be transformed to a simpler lagrangian expressed in terms of spherical polar coordinates. For a spherical top of moment of inertia I = I1 = I2 = I3, the rotational kinetic energy is Eq. (4) multiplied on both sides by I. It is seen that the number of lagrange variables is reduced from three to two, and the Euler Lagrange equations of motion are greatly simplified. Then, gravitational terms can be added as in Horst’s dumbbell representation of the earth in orbit around the sun.


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