This is the usual orbital problem of m attracted by M according to the usual Hooke / Newton inverse square law of force, the concept of force having been proposed by Kepler (Koestler, “The Sleepwalkers” online). The frame (1, 2, 3) is a generalization of the usual frame (plane polar or spherical polar). So in a planar orbit (r, 1, 2, 3) reduces to (r, phi) of the plane polar coordinates, in a three dimensional orbit (r, 1, 2, 3) reduces to (r, theta, phi) of the spherical polar coordinates). So the Eulerian angles are used to define the spin connection matrix in Eq. (10), with omega sub i, i = 1, 2, 3 defined by Eqs. (6) to (8). Therefore the angular velocity expressed in terms of spherical polar coordinates has been re expressed in terms of Eulerian angles as in Note 370(9) and similarly for the spin connection. The Eulerian angles define the rotation from the inertial frame (X, Y, Z) to frame (1, 2, 3). The inertial frame contains only the Newtonian force as you know (no centrifugal or Coriolis forces). In the spherical polar coordinate system r bold = r e sub r, where e sub r is the radial unit vector. In the (1, 2, 3) frame the same r bold is defined as
r bold = r sub 1 e sub 1 + r sub 2 e sub 2 + r sub 3 e sub 3
= r e sub r
= X i + Y j + Z k
Here r bold is the vector joining m and M. So to sum up, everything is expressed in frame (1, 2, 3), giving a huge amount of new information which was of course intractable in the eighteenth century.
To: EMyrone@aol.com
Sent: 23/02/2017 10:44:59 GMT Standard Time
Subj: Re: 371(2): Orbital Theory in Terms of Euler AnglesI am not sure if I understand correctly the physical problem to be solved. You have always to discern between the lab and rotating frames. The Lagrangians (1) and (16) are for the rotating frame. So what is r_1, r_2, r_2 in the rotating frame? So far we have considered situations with centre in the rigid body and with a point outside with fixed distance to the centre of the rigid body (gyro with one point fixed). As soon as we relate this to anohter centre of gravity, we obtain additional vectors from the centre of gravity to the centre of rotating frame (1,2,3). I am not sure if eq.(10) is right in this context. We need the complete coordinate transformation to the lab frame and then can define the kinetic energy plus the roational part which is additionally required because of the rigid body.
Horst
Am 23.02.2017 um 11:02 schrieb EMyrone:
It is shown straightforwardly in this note that the orbits of a mass m around a mass M in the frame (1, 2, 3) of the Euler angles can be found by solving simultaneously six Euler Lagrange equaions in six Lagrange variables. The problem is far too complicated to be solved by hand but can be solved numerically using Maxima to give twenty one new types of orbit described on page 4. There are nutations and precessions in the Euler angles phi, theta and chi. The usual assumption made in orbital theory is that the orbit is planar. Note carefully that the inverse square law is still being used for the force of attraction between m and M, but the Eulerian orbits contain far more information than the theory of planar orbits. The inverse square law does not give precession of a planar orbit as is well known. However the same inverse square law gives rise to many types of nutation and precession in the twenty one types of orbit described by the Euler angles. This isa completely new discovery, and it should be noted carefully that it is based on classical rotational dynamics found in any good textbook. It could have been defined in the eighteenth century, but orbital theory became ossified in planar orbits. This theory is part of ECE2 generally covariant unified field theory because all of rotational dynamics are described by a spin connection of Cartan geometry. Eq. (10) of this note is a special case of the definition of the Cartan covariant derivative. This is another major discovery. An art gallery full of graphics can be prepared based on these results, of greatest interest are plots of the twenty one types of orbit, looking for precessions of a point in the orbit such as the perihelion.