Page:Prinzipien der Dynamik des Elektrons.djvu/5

Hamiltonian principle, we have to resort to the fundamental equations of the dynamics of the electron.

This shall happen now; there, the assumptions concerning motion shall be stated somewhat more generally; we confine ourselves, not on translatory motions, but consider rotations as well. We introduce a coordinate system connected with the already previously mentioned framework, which is rigidly fixed with the electron. We write

${\frac {\partial '{\mathfrak {E}}}{dt}},\ {\frac {\partial '{\mathfrak {H}}}{dt}}$

the temporal changes of the field strengths, assessed from this framework, and related to the axis-intersection fixed in the framework. The two first field equations then assume the simple form:

8)

${\begin{array}{cc}{\frac {1}{c}}{\frac {\partial '{\mathfrak {E}}}{\partial t}}=curl\ {\mathfrak {H}}',&{\mathfrak {H}}'={\mathfrak {H}}-{\frac {1}{c}}[{\mathfrak {vE}}],\\\\-{\frac {1}{c}}{\frac {\partial '{\mathfrak {H}}}{\partial t}}=curl\ {\mathfrak {F}},&{\mathfrak {F}}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {vH}}].\end{array}}$

The form of the field equations makes it near at hand, to emphasize a class of preferred motions, namely such ones, whose field is stationary with respect to that framework. To the "preferred" motions, pure translation belongs, as well as the pure rotation of the spherical electron, eventually also translation connect with rotation around the direction of translatory motion; as regards rotation around an inclined axis with respect to the direction of motion, however, the velocity vector won't possess a fixed location in the framework, and the field won't be stationary with respect to the framework. For preferred motions, ${\tfrac {\partial '{\mathfrak {H}}}{\partial t}}$ vanishes; vector ${\mathfrak {F}}$ (decisive for the inner forces according to (8)) is thus the gradient of a scalar $\varphi$; this will be called "convection potential". The "Lagrangian function" is connected to it by the relation:

9)	$L=W_{m}=W_{e}=-\int \int \int dv{\frac {\varrho \varphi }{2}}.$

(In my earlier report, when considering this formula as a generalization of a formula of ordinary potential theory, I have

$U=\int \int \int dv{\frac {\varrho \varphi }{2}}$

as the "force function" of the electron.)

For "preferred" motions, the momentum can now be derived from the Lagrangian function. So, for pure translation, the equations apply:

9a)	$G={\frac {dL}{dq}}$

9b)	${\frac {d{\mathfrak {G}}}{dt}}={\mathfrak {K}},$

which are denoted in analytical mechanics as first and second line of the Lagrangian equations. Furthermore, the known relation from analytical mechanics apply with respect to energy:

9c)	$W=-L+q\cdot {\frac {dL}{dq}}.$

These formulas are already implicitly contained in my previous report; for the Lagrangian function of the spherical electron, one has to set

10)	$L=-{\frac {3}{5}}{\frac {e^{2}}{a}}\cdot \left({\frac {1-\beta ^{2}}{2\beta }}\right)\ln \left({\frac {1+\beta }{1-\beta }}\right),\ \beta ={\frac {q}{c}}.$

From (10a), (9a) and (7), the formula follows especially for the transverse electromagnetic mass:

10a)	${\begin{array}{c}\mu _{r}=\mu _{0}\cdot {\frac {3}{4}}\cdot \psi (\beta )\\\\\psi (\beta )={\frac {1}{\beta ^{2}}}\left\{\left({\frac {1+\beta ^{2}}{2\beta }}\right)\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right\}\end{array}},$

which experienced such an illuminating confirmation by Kaufmann's experiments.

(It is remarkable, that when one assumes homogeneous surface charge instead of volume charge, the Lagrangian function as well as $\mu _{0}$ obtain the factor ${\tfrac {5}{6}}$, though formula (10a) remains valid.)

Also pure rotation belongs to the "preferred motions". Here it follows

11)	$L=C+{\frac {p\vartheta ^{2}}{2}}$, where $C,p$ are constants.

Thus angular momentum becomes

11a)	$M=p\cdot \vartheta ,$

analogous to a material rigid body; for $p$, the "electromagnetic moment of inertia", one obtains

11b)	$p={\frac {1}{7}}\mu _{0}a^{2},$

while with respect to a sphere homogeneously filled with a material mass $M$, the moment of inertia is $P={\tfrac {2}{5}}\cdot M\cdot a^{2}$

$\left\{\mathrm {With\ surface\ charge\ one\ obtains} \ p={\tfrac {1}{3}}\mu _{0}a^{2},\ P={\tfrac {2}{3}}Ma^{2}\right\}.$

At pure rotation, nothing remarkable occurs. At slow translatory motion, one only commits an error of order $\beta ^{2}$, when one sets ${\mathfrak {G}}=\mu _{0}{\mathfrak {q}},{\mathfrak {M}}=p\vartheta$. Angular forces occur at the passage of cathode rays through inhomogeneous fields. Though the angular force is very small, since the electron's radius $a$ is so small; thus according to my estimation, the energy in the angular motion emerging in the fields of the greatest attainable inhomogeneity,