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* Page found: Das Zweikörperproblem (eq math.1915.76)

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\begin{align}
  & \phi -{{\phi }_{o}}=\int\limits_{{{r}_{o}}}^{r}{\frac{dr\acute{\ }}{r{{\acute{\ }}^{2}}}}\frac{1}{\sqrt{\frac{2mE}{{{l}^{2}}}+\frac{2mk}{{{l}^{2}}r\acute{\ }}-\frac{1}{r{{\acute{\ }}^{2}}}}}=\int\limits_{{{r}_{o}}}^{r}{\frac{dr\acute{\ }}{r{{\acute{\ }}^{2}}}}\frac{1}{\sqrt{D\left[ 1-\frac{1}{D}{{\left( \frac{1}{r{{\acute{\ }}^{{}}}}-\frac{mk}{{{l}^{2}}} \right)}^{2}} \right]}}=\int\limits_{{{r}_{o}}}^{r}{\frac{dr\acute{\ }}{r{{\acute{\ }}^{2}}}}\frac{1}{\sqrt{D}\left[ 1-\frac{1}{D}{{\left( \frac{1}{r{{\acute{\ }}^{{}}}}-\frac{mk}{{{l}^{2}}} \right)}^{{}}} \right]} \\
 & \int\limits_{{{r}_{o}}}^{r}{\frac{dr\acute{\ }}{r{{\acute{\ }}^{2}}}}\frac{1}{\sqrt{D}\left[ 1-\frac{1}{D}{{\left( \frac{1}{r{{\acute{\ }}^{{}}}}-\frac{mk}{{{l}^{2}}} \right)}^{{}}} \right]}=\int\limits_{{{\vartheta }_{0}}}^{\vartheta }{d\vartheta \acute{\ }\sin \vartheta \acute{\ }\frac{1}{\sqrt{1-{{\cos }^{2}}\vartheta }\acute{\ }}=}\int\limits_{{{\vartheta }_{0}}}^{\vartheta }{d\vartheta \acute{\ }=\vartheta -{{\vartheta }_{0}}} \\
 & \vartheta -{{\vartheta }_{0}}=\arccos \frac{1}{\sqrt{D}}\left( \frac{1}{{{r}^{{}}}}-\frac{mk}{{{l}^{2}}} \right)-\arccos \frac{1}{\sqrt{D}}\left( \frac{1}{{{r}_{o}}^{{}}}-\frac{mk}{{{l}^{2}}} \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&\phi -{{\phi }_{o}}=\int \limits _{{r}_{o}}^{r}{\frac {dr{\acute {\ }}}{r{{\acute {\ }}^{2}}}}{\frac {1}{\sqrt {{\frac {2mE}{{l}^{2}}}+{\frac {2mk}{{{l}^{2}}r{\acute {\ }}}}-{\frac {1}{r{{\acute {\ }}^{2}}}}}}}=\int \limits _{{r}_{o}}^{r}{\frac {dr{\acute {\ }}}{r{{\acute {\ }}^{2}}}}{\frac {1}{\sqrt {D\left[1-{\frac {1}{D}}{{\left({\frac {1}{r{{\acute {\ }}^{}}}}-{\frac {mk}{{l}^{2}}}\right)}^{2}}\right]}}}=\int \limits _{{r}_{o}}^{r}{\frac {dr{\acute {\ }}}{r{{\acute {\ }}^{2}}}}{\frac {1}{{\sqrt {D}}\left[1-{\frac {1}{D}}{{\left({\frac {1}{r{{\acute {\ }}^{}}}}-{\frac {mk}{{l}^{2}}}\right)}^{}}\right]}}\\&\int \limits _{{r}_{o}}^{r}{\frac {dr{\acute {\ }}}{r{{\acute {\ }}^{2}}}}{\frac {1}{{\sqrt {D}}\left[1-{\frac {1}{D}}{{\left({\frac {1}{r{{\acute {\ }}^{}}}}-{\frac {mk}{{l}^{2}}}\right)}^{}}\right]}}=\int \limits _{{\vartheta }_{0}}^{\vartheta }{d\vartheta {\acute {\ }}\sin \vartheta {\acute {\ }}{\frac {1}{{\sqrt {1-{{\cos }^{2}}\vartheta }}{\acute {\ }}}}=}\int \limits _{{\vartheta }_{0}}^{\vartheta }{d\vartheta {\acute {\ }}=\vartheta -{{\vartheta }_{0}}}\\&\vartheta -{{\vartheta }_{0}}=\arccos {\frac {1}{\sqrt {D}}}\left({\frac {1}{{r}^{}}}-{\frac {mk}{{l}^{2}}}\right)-\arccos {\frac {1}{\sqrt {D}}}\left({\frac {1}{{{r}_{o}}^{}}}-{\frac {mk}{{l}^{2}}}\right)\\\end{aligned}}

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\begin{matrix} ϕ - ϕ_{o} = \int_{r_{o}}^{r} \frac{d r \overset{´}{}}{r {\overset{´}{}}^{2}} \frac{1}{\sqrt{\frac{2 m E}{l^{2}} + \frac{2 m k}{l^{2} r \overset{´}{}} - \frac{1}{r {\overset{´}{}}^{2}}}} = \int_{r_{o}}^{r} \frac{d r \overset{´}{}}{r {\overset{´}{}}^{2}} \frac{1}{\sqrt{D [1 - \frac{1}{D} {(\frac{1}{r {\overset{´}{}}^{}} - \frac{m k}{l^{2}})}^{2}]}} = \int_{r_{o}}^{r} \frac{d r \overset{´}{}}{r {\overset{´}{}}^{2}} \frac{1}{\sqrt{D} [1 - \frac{1}{D} {(\frac{1}{r {\overset{´}{}}^{}} - \frac{m k}{l^{2}})}^{}]} \\ \int_{r_{o}}^{r} \frac{d r \overset{´}{}}{r {\overset{´}{}}^{2}} \frac{1}{\sqrt{D} [1 - \frac{1}{D} {(\frac{1}{r {\overset{´}{}}^{}} - \frac{m k}{l^{2}})}^{}]} = \int_{ϑ_{0}}^{ϑ} d ϑ \overset{´}{} \sin ϑ \overset{´}{} \frac{1}{\sqrt{1 - \cos^{2} ϑ} \overset{´}{}} = \int_{ϑ_{0}}^{ϑ} d ϑ \overset{´}{} = ϑ - ϑ_{0} \\ ϑ - ϑ_{0} = \arccos \frac{1}{\sqrt{D}} (\frac{1}{r^{}} - \frac{m k}{l^{2}}) - \arccos \frac{1}{\sqrt{D}} (\frac{1}{{r_{o}}^{}} - \frac{m k}{l^{2}}) \end{matrix}

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data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>k</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>l</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><munderover><mo form="prefix" stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>o</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>r</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>r</mi><msup><mover><mtext>&#160;</mtext><mo 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data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>k</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>l</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo><munderover><mo form="prefix" stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>ϑ</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>ϑ</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>ϑ</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mi>sin</mi><mo>&#x2061;</mo><mi>ϑ</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow 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stretchy="false">−</mo><msub><mi>ϑ</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">=</mo><mi>arccos</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msqrt><mi>D</mi></msqrt></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mi>r</mi><mrow data-mjx-texclass="ORD"></mrow></msup></mrow></mfrac></mrow><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>k</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>l</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">−</mo><mi>arccos</mi><mo>&#x2061;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msqrt><mi>D</mi></msqrt></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>o</mi></mrow></msub><mrow data-mjx-texclass="ORD"></mrow></msup></mrow></mfrac></mrow><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>k</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>l</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$ϕ$

$ϕ_{o}$

$r_{o}$

$r$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$m$

$E$

$l$

$m$

$k$

$l$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$r_{o}$

$r$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$D$

$D$

$r$

$\overset{´}{}$

$m$

$k$

$l$

$r_{o}$

$r$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$D$

$D$

$r$

$\overset{´}{}$

$m$

$k$

$l$

$r_{o}$

$r$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$D$

$D$

$r$

$\overset{´}{}$

$m$

$k$

$l$

$ϑ_{0}$

$ϑ$

$ϑ$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$ϑ_{0}$

$ϑ$

$ϑ$

$\overset{´}{}$

$ϑ$

$ϑ_{0}$

$ϑ$

$ϑ_{0}$

$D$

$r$

$m$

$k$

$l$

$D$

$r_{o}$

$m$

$k$

$l$

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