Page:Grundgleichungen (Minkowski).djvu/45

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(92) N_{h}=-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\epsilon}{\partial x_{h}}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial x_{h}}

+(\epsilon\mu-1)\left(\Omega_{1}\frac{\partial w_{1}}{\partial x_{h}}+\Omega_{2}\frac{\partial w_{2}}{\partial x_{h}}+\Omega_{3}\frac{\partial w_{3}}{\partial x_{h}}+\Omega_{4}\frac{\partial w_{4}}{\partial x_{h}}\right)

for h=1,2,3,4.

Now if we make use of (59), and denote the space-vector which has \Omega_{1},\ \Omega_{2},\ \Omega_{3} as the x-, y-, z-components by the symbol \mathfrak{W}, then the third component of 92) can be expressed in the form

(93) \frac{\epsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{d\mathfrak{w}}{dx_{h}}\right)

The round bracket denoting the scalar product of the vectors within it.

§ 14. The Ponderomotive Force.

Let us now write out the relation K = lor\ S = -sF + N in a more practical form; we have the four equations

(94) K_{1}=\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial X_{t}}{\partial t}=\varrho\mathfrak{E}_{x}+\mathfrak{s}_{y}\mathfrak{M}_{z}-\mathfrak{s}_{z}\mathfrak{M}_{y}

-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\epsilon}{\partial x}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial x}+\frac{\epsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial x}\right),

(95) K_{2}=\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial Y_{t}}{\partial t}=\varrho\mathfrak{E}_{y}+\mathfrak{s}_{z}\mathfrak{M}_{x}-\mathfrak{s}_{x}\mathfrak{M}_{z}

-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\epsilon}{\partial y}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial y}+\frac{\epsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial y}\right),

(96) K_{3}=\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial Z_{t}}{\partial t}=\varrho\mathfrak{E}_{z}+\mathfrak{s}_{x}\mathfrak{M}_{y}-\mathfrak{s}_{y}\mathfrak{M}_{x}

-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\epsilon}{\partial z}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial z}+\frac{\epsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial z}\right),

(97) \frac{1}{i}K_{4}=-\frac{\partial T_{x}}{\partial x}-\frac{\partial T_{y}}{\partial y}-\frac{\partial T_{z}}{\partial z}-\frac{\partial T_{t}}{\partial t}=\mathfrak{s}_{x}\mathfrak{E}_{x}+\mathfrak{s}_{y}\mathfrak{E}_{y}+\mathfrak{s}_{z}\mathfrak{E}_{z}

+\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\epsilon}{\partial t}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial t}+\frac{\epsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial t}\right).

It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point x,y,z,t, it has got x-, y-, z-