Page:AbrahamMinkowski2.djvu/13

form^[1]

(32)

${\begin{cases}\nu {\frac {d^{2}x}{d\tau ^{2}}}={\mathfrak {K}}_{x}+\Psi {\frac {dx}{cd\tau }},\\\\\nu {\frac {d^{2}y}{d\tau ^{2}}}={\mathfrak {K}}_{y}+\Psi {\frac {dy}{cd\tau }},\\\\\nu {\frac {d^{2}z}{d\tau ^{2}}}={\mathfrak {K}}_{z}+\Psi {\frac {dz}{cd\tau }},\\\\\nu {\frac {d^{2}u}{d\tau ^{2}}}={\mathfrak {K}}_{u}+\Psi {\frac {du}{cd\tau }},\end{cases}}$

where $S^{4}(\nu )$ determines the "rest density" of matter. The identity (30a), from which it follows

${\frac {dx}{d\tau }}{\frac {d^{2}x}{d\tau ^{2}}}+{\frac {dy}{d\tau }}{\frac {d^{2}y}{d\tau ^{2}}}+{\frac {dz}{d\tau }}{\frac {d^{2}z}{d\tau ^{2}}}+{\frac {du}{d\tau }}{\frac {d^{2}u}{d\tau ^{2}}}=0$

is satisfied by equations (32).

Minkowski denotes the ponderomotive force of the electromagnetic field, i.e. the $V^{3}$ whose components are the second members of the first three equations of motion (32), i.e. the $V^{3}$, by:

(32a)

${\mathfrak {K}}+\Psi {\mathfrak {q}}k^{-1}={\mathfrak {K}}-{\frac {{\mathfrak {q}}\cdot Q}{ck^{2}}}$

This vector is not identical to the force determined from the momentum theorem (14), but it differs from that by

$-{\frac {{\mathfrak {q}}\cdot Q}{ck^{2}}}$

Thus, when the Joule-heat emerges in matter, then the mechanics of Minkowski must add this additional force to the ponderomotive force, which is derived from the momentum theorem.

Considering the momentum theorem as being important to electromagnetic mechanics, I prefer to keep this principle of the electrodynamics of moving bodies. We can remove the additional force of Minkowski, by giving the equations of motion, instead of (32), precisely the form suggested by the mechanical law of momentum:

(33)

${\begin{cases}{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)={\mathfrak {K}}_{x},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)={\mathfrak {K}}_{y},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {dz}{d\tau }}\right)={\mathfrak {K}}_{z},\\\\{\frac {d}{d\tau }}\left(\nu {\frac {du}{d\tau }}\right)={\mathfrak {K}}_{u}.\end{cases}}$

Since $\tau$ and $\nu$ are of $S^{4}$, both members of these equations are the components of $V_{I}^{4}$; then these equations agree with the principle of relativity. The identity

1. H. Minkowski, l. c. equation (20), pag. 54.