## Further Discussion of Note 371(3) and 371(2)

The point mass M is placed at the origin and the point mass m is at (r1, r2, r3). The motion of the axes e1, e2, and e3 is defined with the dynamics of the Euler angles through the spin connection and equations (6) to (8). This gives six equations in six unknowns, Eqs. (17) to (22). That is an exactly determined problem therefore. The three degrees of freedom are the three dimensions of three dimensional space. Only three coordinates are being used, e sub 1, e sub 2 and e sub 3, and six Lagrange variables, r1, r2, r3, theta, phi and chi. The Euler equations for a rigid object are not being used, because there are no moments of inertia being used. Have you tried running these six simultaneous equations through Maxima, to find how the system behaves? If it gives reasonable results all looks OK. In spherical polar coordinates the mass M is at the origin and the mass m is at vector r. Solutions in the spherical polar system are also full of interest and much less complicated. It is also possible to use your idea of a unit vector by adapting the gyroscope with one point fixed. There are many interesting things to work on adn they will all create a lot of interest. It is best to work with spherical polar coordinates by Ockham’s Razor, but the Euler angles give a very large amount of new information. The important thing is the ability of Maxima so solve very complicated sets of simultaneous differential equations.

To: EMyrone@aol.com
Sent: 25/02/2017 09:09:39 GMT Standard Time
Subj: Re: Discussion of Note 371(3) and 371(2)

Your argument on time-dependence of the (e1, e2, e3) frame is correct. However it is not possble to describe a probelm with 3 degreees of freedom by 6 coordinates, at least not in Lagrange theory. Then you obtain an underdetermined system of equations, there is no unique solution.
A second point is to strictly discern if a masspoint or a rigid body is considered. You cannot use the Lagrange equations for a rigid body and apply it to a masspoint because then you have too many coordinates.
On the other hand it is possible to use Eulerian angles for masspoints. However you have already 3 coordinates so you cannot introduce additional translations. This type of application seems to be restricted to pure rotations on a unit sphere. The situation is different for a rigid body again.

Horst

Am 25.02.2017 um 09:53 schrieb EMyrone:

It is a good idea to use the gyro with one point fixed for orbital theory, the mass M is at the fixed point of the gyro, and mass m is separated by a distance r from M. Unlike problem 10.10 of Marion and Thornton, however, the distance r is not constant. I will look in to this and go back to the basics of the derivation of the Euler equation from variational calculus, Marion and Thornton chapter five (Euler 1744). Howevber, I think that all is OK for the following reasons. It is true that the Euler angles relating frame (i, j, k) and (e1, e2, e3) are constants by definition, provided that frame (e1, e2, e3) always has the same orientation with respect to frame (i, j, k) and provided that the two frames are static. Then theta, phi and chi, being constants, cannot be used as variables. I agree about this point. However in Note 371(2), frame (e1, e2, e3) is moving with respect to (i. j. k), and so theta, phi and chi are also moving. This is because the Cartesain frame (i, jk, k) is static by definition, but frame (e1, e2, e3) is dynamic, i.e. e1, e2 and e3 depend on time, but i, j, and k do not depend on time. Similarly in spherical polars, (i, j, k) static, but (e sub r, e sub theta, e sub phi) is time dependent so r, theta and phi all depend on time. The lagrangian (1) of Note 371(2) is true for any definition of v, and Eq. (10) of that note is true for any definition of the spin connection (e.g. plane polar, spherical polar, Eulerian, and any curvilinear coordinate system in three dimensions). So Eqs. (11) to (16) of that note are correct. So it is correct to set up the lagrangian (16) using the Lagrange variables r1(t), r2(t), r3(t), theta(t), phi(t) and chi(t). It could also be set up with plane polar or spherical polar coordinates. We have already correctly solved those problems using the lagrangian method, The fundamental property of the spin connection is to show how the axes themselves move and it is valid to re express the angles of the plane polar and spherical polar coordinates as Eulerian angles. In the orbital problem, the Eulerian angles are all time dependent. So they vary in this sense, and can be used as Lagrange variables. The Euler variable x of chapter five of Marion and Thornton is t, x = t. To sum up, it is true that the Euler angles are constants when viewed as angles defining the orientation of a static (1, 2, 3) with respect to a static (X, Y, Z), but in Eq. (10) of Note 371(2), the components of the spin conenction are defined in terms of time dependent Euler angles, which are therefore Lagrange variables. This fact can be seen from Eqs. (6) to (8) of the note, in which appear phi dot, theta dot, and chi dot. These are in general non zero, i.e. they are all time dependent angular velocities.

To: EMyrone
Sent: 24/02/2017 18:35:19 GMT Standard Time
Subj: Re: Note 371(3) : Definition of Reference Frames

Thanks, this clarifies the subject. I think we must be careful in applying Lagrange theory. A mass point in 3D is described by 3 variables that are either [X. Y. Z] or [r1, r2, r3]. The Eulerian angles describe the coordinate transformation between both frames of reference bold [i, j, k] and bold [e1, e2, e3]. The Eulerian angles are the sam for all points [r1, r2, r3]. So they cannot be subject to variation in the Lagrange mechanism. The degree of freedom must be the same in both frames, otherwise we are not dealing with generalized coordinates and Lagrange theory cannot be applied.

A Lagrange approach in Eulerian coordinates can be made if we describe the motion of a mass point that is described as a unit vector in the [e1, e2, e3] frame. Then [r1, r2, r3] is fixed and the Eulerian angles can indeed be used for Lagrange variation. This is done for the gyro with one point fixed.

I calculated the transformation matrix A of the note and its inverse. The order of rotations is different from that in M&T.

Horst

Am 24.02.2017 um 15:11 schrieb EMyrone:

In this note the reference frames used in Note 371(2) are defined. The mass m orbits a mass M situated at the origin. The spherical polar coordinates are defined, and frame (X, Y, Z) is rotated into frame (1, 2, 3) with the matrix of Euler angles as in Eq. (11). The inverse of this matrix can be used to define r1, r2, and r3 in terms of X, Y, Z. In plane polar coordinates the orbit is a conic section with M at one focus as is well known.The planar elliptical orbit does not precess, but using spherical polar coordinates and Eulerian angles the orbit is no longer planar and precesses.