## Sketch of the Dumbbell Model by Horst Eckardt

This is most helpful and many thanks! It can be written up in UFT369 or UFT370, and is an incisive way of explaining Milankovitch cycles.

To: EMyrone@aol.com
Sent: 13/02/2017 13:49:00 GMT Standard Time
Subj: Re: First Results for the Milankovitch Cycles

See sketch: The model is two dumbbell masses moving freely around the middle point (centre of mass) of their connection (angles theta and phi). Their distance to the middle point is h=const. The middle point moves with coordinates r, theta_1, phi_1 around the central mass M (at rest). A So this is a two-body problem with vectorial coordinates

r_1 = R + h_1,
r_2 = R + h_2.

Since

h_1 = – h_2

the kinetic energies of both masses are the same and the total kinetic energy takes a simple form. If the correct distances to the centre are taken for the potential energy, the calculation becomes unhandable, therefore I took the simplified form

U = – m M G / r

for both masses. In total we have 5 degrees of freedom or independent Lagrange equations. They are compiled in section 6 of the protocol. The subsystems (theta, phi) and (r, theta_1, phi_1) decouple. The solutions are oscillatory in the parameter range where the conic section of the centre of mass is an ellipse. The exact term of U would introduce a coupling between both subsystems.
The dumbbell without central motion should be equivalent to a rigid body with special moments of inertia.

Horst

Am 13.02.2017 um 13:29 schrieb EMyrone:

This looks very interesting, the problem has been essentially solved. Can you send over the lagrangian used for this solution? It will be very interesting. The result is right, the nutations and precessions are very slow compared with the orbital interval. Congratulations on this solution!

To: EMyrone
Sent: 12/02/2017 19:36:52 GMT Standard Time
Subj: dumbbell orbit

I plotted the orbit of the centre of the dumbbell (blue) and one mass of
the dumbbell (red). It can be seen that the centre of mass moves on a
plane elliptic orbit. The dumbbell shows nutation and precession. This
is the most simple model without a rigid body, requiring two mass points
with constraints. These mass points represent a ring which gives the
same Lagrange equations. This may also be a method for computing the
Milankovitch cycles where the parameters are different:
nutation/precession is very slow compared to the elliptic orbit.

Horst

Lagrange-dumbbell.pdf