370(9): Orbital Kinetic Energy and Lagrangian

Thie note expresses any three dimensional orbital motion of m around M in terms of the orbital spherical polar coordinates, Eqs. (18) to (22), and in terms of orbital Euler angles, Eqs. (24) to (28). These equations give the nutations and precessions of the centre of mass of the orbiting object. A three dimensional orbit is much more intricate than a two dimensional orbit. In addition to the orbital kinetic energy there is a spin kinetic energy due to the internal nutations and precessions of the object in orbit. This internal or rotational kinetic energy is 1/2 sigma I sub i omega squared sub i, i = 1, 2, 3, where I sub i are the three principal moments of inertia and omega sub i the angular velocities in the internal frame of the object (e.g. the planet Earth in orbit around the sun). Either the spherical polar coordinates or the Euler angles can be used. In the former case tehre are five Lagrange variables in general, three for the orbit, and two for the rotational kinetic energy. In the latter case there are seven Lagrange variables, four for the orbit and three for the internal or rotational kinetic energy. If the orbit is considered to be planar the number of Lagrange variables is reduced because plane polar coordinates can be used for the orbit. These sets of simultaneous differential equations can be solved by Maxima, giving all the information needed for any problem, for example the Milankovitch cycles, and a gyroscope subjected to an external force or torque. It is matter of setting up and defining the lagrangian, and identifying the Lagrange variables. In principle this is straightforward, and the Maxima code gives all the solutions in tems of various trajectories. This is the great advantage of the code.


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