Many thanks, this is full of interest and the model looks like a valid approximation of the nutations and the precessions of the earth in orbit around the sun. This formidable problem has been solved, and consists of five simultaneous equations. There are regular spikes in the results, and this method gives an entirely new insight into orbital motion. I notice from feedback today that there is already intense interest in UFT368 and UFT369. These results could go into your Section of UFT369.
Sent: 09/02/2017 16:59:26 GMT Standard Time
Subj: Re: Dumbbell simulation
The equations for the dumbbell on a gravitational orbit can only be handled when the potential energy is approximated by
U = – mMG/r
where r is the distance of the dumbell centre to the gravitational centre. It is difficult to find initial conditions with stable solutions. In the example the dumbbell comes very near to the centre (see r(t) in last figure). There the approximation for U is no more valid. I used two sets of spherical coordinates for the dumbbell and the centre of motion
theta – dumbbell
theta_1 – centre of motion
These are 5 independent coordinates. It comes out that the motion stays in a plane, although the dumbbell rotates in the theta direction (polar angle). The azimutahl angles are phi and phi_1 as usual.
Obviously such a construct can only produce coupling between centre and dumbbell coordinates if gravitational forces depend on the position of the dumbbell masses (not assumed here).
Am 09.02.2017 um 12:40 schrieb EMyrone:
Many thanks, very interesting! As requested I have just sent over the results for a gyroscope subjected to a force applied directly against the force of gravity to the point of the gyro, so that it lifts the point of the gyro. I suggest that we start the study of the Milankovitch cycles with the lagrangian (2) of Note 370(1). This will give independent rotational and translational motions of the earth, but to start with, these will be very interesting in themselves. An asymmetric top model can be used in general, in which there are three principal moments of inertia. Then I can begin to explore the use of a non central potential, Eq. (16) of Note 370(1), the idea being to find a way of dealing with the influence of the orbital motion of the earth on its own nutations and precessions. I think that this is what the complete Milankovitch obervations needs for their description.
Sent: 09/02/2017 11:02:48 GMT Standard Time
Subj: dumbbell simulation
It seems that the coordinates of the centre of mass and the dumbell
rotation decouple. This could be the reason for the strange results of
the moments of inertia approach. Will send over the results in the