Discussion of Note 369(9)

Rotational motion of the earth about its own centre of mass (nutation and precession) is introduced by defining the position vector in frame (1, 2, 3) according to Eq. (5). This is the position of any point in the asymmetric top with respect to its centre of mass. Frame (1, 2, 3) is rotating with respect to the lab frame (X, Y, Z) and the rotational Euler transformation (6) relates the unit vectors in frame (1, 2, 3) to the unit vectors in frame (X, Y, Z). This Euler transformation results in equation (9), which defines r1 in frame (X, Y, Z), so that it can be vector summed to r, the distance from the centre of mass to the origin of frame (X, Y, Z). In a planar orbit around the sun, the centre of mass is at one focus of an ellipse as usual. The results would be very interesting because they would define the nutations and precessions of the earth at any point in its orbit around the sun. The rotational motion of the earth about its own centre of mass is described by the Euler angles theta, phi, chi, and the translational motion of its centre of mass in a planar orbit around the sun is described by the plane polar coordinates r and theta1. The method inter relates all five lagrange variables: r, theta1, theta, phi and chi. This could never be solved by hand without immediately introducing drastic approximations which lose information.

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