# zbMATH — the first resource for mathematics

Republication of: New mechanics of material systems. Translation from [Acta Phys. Polon. 6, 163–200 (1937)]. (English) Zbl 1188.83018
Summary: “We assume the laws of the field (electromagnetic and gravitational) outside matter in the form of the Maxwell-Einstein equations. (Our deductions arc, anyway, based only on certain fundamental attributes of the structure of those equations.) We characterize matter through the potentials of the field that it creates. These potentials can be represented as sums of potentials of multipoles (similarly to the Newtonian potential of a matter lump, outside the lump). In this way we arrive at the notion of the gravitational skeleton of a charge-free material system. In a gravitational skeleton, a single pole characterizes the mass, the dipole and the quadrupole – the rotation moment. This assignment is mutually unique. From the equations obeyed by the potentials in the four-dimensional world we obtain the equations of mechanics in the form of equations with ordinary derivatives that determine the motion of the center of mass of the system and the changes of our multipoles with time. From these equations it follows that the motion of the center of mass and the rotation are connected to each other. The new terms in the equations of mechanics that cause this connection vanish if it is assumed that the velocity of light is infinitely large. The energy equation for the new equations of mechanics contains a new term: the acceleration energy.
Taking into account the multipoles, up to the quadrupole, is necessary and sufficient to obtain the laws of motion that go over into the laws of classical mechanics of a system of points and of a rigid body, when it is assumed that the velocity of light is infinitely large. The interpretation of the multipoles via classical quantities is approximate, but the equations of motion obtained for them are a mathematically exact condition of existence of multipole singularities in the gravitational solutions outside matter. Therefore, one can consider the multipoles to be an independent dynamical characteristic of the system.
The rotation axis is present here as a secondary construction (and this is a fundamental issue), while the quantity that primarily characterizes rotation is an antisymmetric tensor. We arrive at the connection between rotation and the motion of the center of mass according to a law which differs from Fokker’s assumption about the motion of the axis of a symmetric top.”
An editorial note to this paper can be found in this issue preceding this Golden Oldie [Gen. Relativ. Gravitation 42, No. 4, 985–987 (2010; Zbl 1188.83006)]. The companion paper to this one can be found in this issue preceding this Golden Oldie [Gen. Relativ. Gravitation 42, No. 4, 989–1010 (2010; Zbl 1188.83017)].
Original paper: M. Mathisson, “Neue Mechanik materieller Systeme”, Acta Phys. Polon. 6, 163–200 (1937; Zbl 0017.43006). Reprinted with the kind permission of the Editors of Acta Physica Polonica. Translated by Anita Ehlers.

##### MSC:
 83C10 Equations of motion in general relativity and gravitational theory 83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
Full Text: