Further Discussion of Note 371(3)

This is a very good idea, many ideas such as this can be developed.

To: EMyrone@aol.com
Sent: 25/02/2017 07:21:27 GMT Standard Time
Subj: PS: Re: Note 371(3) : Definition of Reference Frames

PS: a solution can be to consider a masspoint with 3 internal degrees of freedom, corresponding to the Euler angles. In addition there are 3 degrees of freedom for external motion (cartesian or spherical coordinate system). The internal degrees of freedom represent a spin resp. spin moment. It is not useful to introduce further translational coordinates arbitrarily on the surface of a body or so, because this does not conform to the Lagrange concept. The latter is made for masspoints only.

Horst

Am 24.02.2017 um 19:34 schrieb Horst Eckardt:

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.