Popper: Equations of motion of rigid bodies

SYNOPSIS: The problem of setting up the equations of the motion of rigid bodies seems to have entirely solved. However, even Foeppl did find it necessary to remark that, in case of complicated composite motions, setting up the equations is difficult and requires the invention of a new way for each individual problem.

Although widely applicable, the Lagrange formulas do not fit non-holonomic conditions. Using them one can neither reckon with nor discover forces doing no actual work in the system. Moreover, one cannot make use of electronic analog computers as derivatives have to be made with respect to the coordinates and not to time only.

This paper offers a formula (No. 40 in the text) eliminating all the above difficulties.

As a starting point we take Newton's law for an elementary mass of a rigid body (say
.
_ _
P=m v ),
without the aid of the moment of momentum, the latter being limited to special applications only.

The formula (40) is presented in three rows. The first consists of the Euler equations, and the third includes only expressions resulting of the rotational movements when the main axes of moment of inertia of the body do not coincide with the axes of the coordinate system. The second row (which has more members than the two others put together) includes expressions resulting from the translational movements and the combination of the same with rotation.

Former attempts have been made to develop similar formulae. Noteworthy is the formula developed by Prof. Deimel⁽²⁾. Deimel obtains (in his terms) the first and the third rows of our formula (40). Unfortunately, he was completely mistaken with the second row.

After having developed [in this paper] the main formula (40), its field of application is discussed theoretically and in practice by means of technical examples.

ABSTRACT IN HEBREW

6 references:
2. Deimel, R.F., "Mechanics of the gyroscope; The dynamics of rotation," Dover, New York (1950).
- 1st edition: Macmillan (1929).

and 5 more references.

COMMENT: Popper used in equations an overline for a vector and an over dot for derivative with respect to the time. These signs might be placed incorrectly in some browsers.

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