Agreed with the first remark. They fix the position of a point on the earth’s surface relative to its centre of mass, and this also answers the second point, because they define the principal moments of inertia (Marion and Thornton chapter ten of the third edition). Note 369(9) was a development of your earlier suggestion of using Cartesian coordinates in the lab frame to develop all the possible interactions. Introducing a rotational lagrangian is superfluous because rotational motion is already included through the motion of vector r sub 1. As you can see this was initially defined in the frame (1, 2, 3) of the principal moments of inertia of the gyroscope (the earth) and then transformed into (X, Y, Z). The Euler transformation is inevitably very complicated but the idea is simple. The great advantage of this method, if it can be crunched out, is that it accounts for all possible motions with five Lagrange variables, r, theta1, theta, phi, chi.
Sent: 05/02/2017 15:46:41 GMT Standard Time
Subj: Re: 369(9): General Theory of the Milankovitch Cycles
I have some difficulties of understanding:
1. The constants r11, r12, r13 describe the position of an arbitrary point in the rotating body. Can these be chosen arbitrarily so that we can observe the motion of a point at the earth surface for example?
2. In the Lagrangian (11) there are no rotational energy terms. Therefore the Lagrange equations will not contain any moments of inertia. I guess that these terms of kinetic energy are to be added and you omitted them for clarity.
Am 04.02.2017 um 13:15 schrieb EMyrone:
This general theory shows that five simultaneous Euler Lagrange equations in five Lagrange variables must be solved in general. All the motions are inter related, and are made up of the nutations and precessions of an asymmetric top in orbit around the sun. So the Milankowitch cycles are very intricate. It may be possible to solve this problem numerically using Maxima’s code for simultaneous differential equations. The nutations and precessions depend on the position of the orbit. For example they are different at perihelion and aphelion. The earth is approximately a symmetric top and one of its precessions results in the equinox precession developed in UFT119 with the complementary method of the gravitomagnetic field. So if possible, and when Horst has time, graphics of these intricate motions would be full of interest.