Editing Relativistisches Hamiltonprinzip
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Die rel. Dynamik eines Massepunktes kann aus dem Extremalprinzip abgeleitet werden, wenn man Die Punkt 1 und 2 als Anfangs- und Endereignis im 4- Raum sieht und wenn man die Ränder bei Variation festhält: | Die rel. Dynamik eines Massepunktes kann aus dem Extremalprinzip abgeleitet werden, wenn man Die Punkt 1 und 2 als Anfangs- und Endereignis im 4- Raum sieht und wenn man die Ränder bei Variation festhält: | ||
<math>\begin{align} | |||
& \delta W=0 \\ | & \delta W=0 \\ | ||
& W=\int_{1}^{2}{{}}ds \\ | & W=\int_{1}^{2}{{}}ds \\ | ||
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letzteres: Wirkungsintegral | letzteres: Wirkungsintegral | ||
Wichtig: | Wichtig: | ||
<math>{{\left. \delta {{x}^{i}} \right|}_{1,2}}=0</math> | |||
Newtonsche Mechanik ist Grenzfall: | Newtonsche Mechanik ist Grenzfall: | ||
<math>W=-{{m}_{0}}c\int_{1}^{2}{{}}ds</math> | |||
Wechselwirkung eines Massepunktes mit einem 4- Vektor- Feld | Wechselwirkung eines Massepunktes mit einem 4- Vektor- Feld | ||
<math>\begin{align} | |||
& \left( {{\phi }^{i}} \right)({{x}^{j}}) \\ | & \left( {{\phi }^{i}} \right)({{x}^{j}}) \\ | ||
& \Rightarrow \\ | & \Rightarrow \\ | ||
\end{align}</math> | \end{align}</math> | ||
<math>W=\int_{1}^{2}{{}}\left\{ -{{m}_{0}}cds-{{\phi }^{i}}d{{x}_{i}} \right\}</math> | |||
mit den Lorentz- Invarianten | mit den Lorentz- Invarianten | ||
<math>{{m}_{0}}cds</math> | |||
und | |||
<math>{{\phi }^{i}}d{{x}_{i}}</math> | |||
'''Variation:''' | '''Variation:''' | ||
<math>\delta W=\int_{1}^{2}{{}}\left\{ -{{m}_{0}}c\delta \left( ds \right)-\delta \left( {{\phi }^{\mu }}d{{x}_{\mu }} \right) \right\}</math> | |||
Nun: | Nun: | ||
<math>\begin{align} | |||
& \delta \left( ds \right)=\delta {{\left( d{{x}^{\mu }}d{{x}_{\mu }} \right)}^{\frac{1}{2}}}=\frac{1}{2}\frac{\left( d\delta {{x}^{\mu }} \right)d{{x}_{\mu }}+d{{x}^{\mu }}\left( d\delta {{x}_{\mu }} \right)}{ds} \\ | & \delta \left( ds \right)=\delta {{\left( d{{x}^{\mu }}d{{x}_{\mu }} \right)}^{\frac{1}{2}}}=\frac{1}{2}\frac{\left( d\delta {{x}^{\mu }} \right)d{{x}_{\mu }}+d{{x}^{\mu }}\left( d\delta {{x}_{\mu }} \right)}{ds} \\ | ||
& \left( d\delta {{x}^{\mu }} \right)d{{x}_{\mu }}=d{{x}^{\mu }}\left( d\delta {{x}_{\mu }} \right) \\ | & \left( d\delta {{x}^{\mu }} \right)d{{x}_{\mu }}=d{{x}^{\mu }}\left( d\delta {{x}_{\mu }} \right) \\ | ||
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Außerdem: | Außerdem: | ||
<math>\delta \left( {{\phi }^{\mu }}d{{x}_{\mu }} \right)=\delta {{\phi }^{\mu }}d{{x}_{\mu }}+{{\phi }^{\mu }}d\left( \delta {{x}_{\mu }} \right)</math> | |||
Somit: | Somit: | ||
<math>\delta W=\int_{1}^{2}{{}}\left\{ -{{m}_{0}}c{{u}^{\mu }}\left( d\delta {{x}_{\mu }} \right)-\delta {{\phi }^{\mu }}d{{x}_{\mu }}-{{\phi }^{\mu }}d\left( \delta {{x}_{\mu }} \right) \right\}</math> | |||
Weiter mit partieller Integration: | Weiter mit partieller Integration: | ||
<math>\begin{align} | |||
& \int_{1}^{2}{{}}-{{m}_{0}}c{{u}^{\mu }}d\left( \delta {{x}_{\mu }} \right)=\left[ -{{m}_{0}}c{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \right]_{1}^{2}+\int_{1}^{2}{{}}{{m}_{0}}cd{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \\ | & \int_{1}^{2}{{}}-{{m}_{0}}c{{u}^{\mu }}d\left( \delta {{x}_{\mu }} \right)=\left[ -{{m}_{0}}c{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \right]_{1}^{2}+\int_{1}^{2}{{}}{{m}_{0}}cd{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \\ | ||
& \left[ -{{m}_{0}}c{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \right]_{1}^{2}=0,weil\delta {{x}_{\mu }}_{1}^{2}=0 \\ | & \left[ -{{m}_{0}}c{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \right]_{1}^{2}=0,weil\delta {{x}_{\mu }}_{1}^{2}=0 \\ | ||
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Weiter: | Weiter: | ||
<math>\int_{1}^{2}{{}}-{{\phi }^{\mu }}d\left( \delta {{x}_{\mu }} \right)=-\left[ {{\phi }^{\mu }}\delta {{x}_{\mu }} \right]_{1}^{2}+\int_{1}^{2}{{}}d{{\phi }^{\mu }}\left( \delta {{x}_{\mu }} \right)</math> | |||
Mit | Mit | ||
<math>\begin{align} | |||
& d{{\phi }^{\mu }}={{\partial }^{\nu }}{{\phi }^{\mu }}d{{x}_{\nu }}={{\partial }^{\nu }}{{\phi }^{\mu }}{{u}_{\nu }}ds \\ | & d{{\phi }^{\mu }}={{\partial }^{\nu }}{{\phi }^{\mu }}d{{x}_{\nu }}={{\partial }^{\nu }}{{\phi }^{\mu }}{{u}_{\nu }}ds \\ | ||
& \delta {{\phi }^{\mu }}={{\partial }^{\nu }}{{\phi }^{\mu }}\delta {{x}_{\nu }} \\ | & \delta {{\phi }^{\mu }}={{\partial }^{\nu }}{{\phi }^{\mu }}\delta {{x}_{\nu }} \\ | ||
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Einsetzen in | Einsetzen in | ||
<math>\delta W=\int_{1}^{2}{{}}\left\{ -{{m}_{0}}c{{u}^{\mu }}\left( d\delta {{x}_{\mu }} \right)-\delta {{\phi }^{\mu }}d{{x}_{\mu }}-{{\phi }^{\mu }}d\left( \delta {{x}_{\mu }} \right) \right\}</math> | |||
liefert: | liefert: | ||
<math>\delta W=\int_{1}^{2}{{}}\left\{ {{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}-\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right){{u}_{\nu }} \right\}\delta {{x}_{\mu }}</math> | |||
'''Wegen''' | '''Wegen''' | ||
<math>\begin{align} | |||
& \delta W=\int_{1}^{2}{{}}\left\{ {{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}-\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right){{u}_{\nu }} \right\}\delta {{x}_{\mu }}=0 \\ | & \delta W=\int_{1}^{2}{{}}\left\{ {{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}-\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right){{u}_{\nu }} \right\}\delta {{x}_{\mu }}=0 \\ | ||
& {{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}=\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right){{u}_{\nu }}:={{f}^{\mu \nu }}{{u}_{\nu }} \\ | & {{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}=\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right){{u}_{\nu }}:={{f}^{\mu \nu }}{{u}_{\nu }} \\ | ||
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Man setze: | Man setze: | ||
<math>\begin{align} | |||
& {{p}^{\mu }}={{m}_{0}}c{{u}^{\mu }} \\ | & {{p}^{\mu }}={{m}_{0}}c{{u}^{\mu }} \\ | ||
& {{f}^{\mu \nu }}=\frac{q}{c}{{F}^{\mu \nu }}=\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right) \\ | & {{f}^{\mu \nu }}=\frac{q}{c}{{F}^{\mu \nu }}=\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right) \\ | ||
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Man bestimmt die Ortskomponenten | Man bestimmt die Ortskomponenten | ||
<math>\alpha =1,2,3</math> | |||
über | über | ||
<math>\begin{align} | |||
& \frac{d}{dt}\bar{p}=q\left( \bar{E}+\bar{v}\times \bar{B} \right) \\ | & \frac{d}{dt}\bar{p}=q\left( \bar{E}+\bar{v}\times \bar{B} \right) \\ | ||
& \\ | & \\ | ||
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überein, denn mit | überein, denn mit | ||
<math>\begin{align} | |||
& {{u}^{0}}=\gamma \\ | & {{u}^{0}}=\gamma \\ | ||
& {{u}^{\alpha }}=\frac{\gamma }{c}{{v}^{\alpha }}=-{{u}_{\alpha }} \\ | & {{u}^{\alpha }}=\frac{\gamma }{c}{{v}^{\alpha }}=-{{u}_{\alpha }} \\ | ||
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folgt dann: | folgt dann: | ||
<math>\begin{align} | |||
& \frac{d}{dt}{{p}^{1}}=q\left( {{E}^{1}}+{{v}^{2}}{{B}^{3}}-{{v}^{3}}{{B}^{2}} \right) \\ | & \frac{d}{dt}{{p}^{1}}=q\left( {{E}^{1}}+{{v}^{2}}{{B}^{3}}-{{v}^{3}}{{B}^{2}} \right) \\ | ||
& =q\left( {{F}^{10}}+{{F}^{21}}\frac{1}{c}{{v}^{2}}-{{F}^{13}}\frac{1}{c}{{v}^{3}} \right) \\ | & =q\left( {{F}^{10}}+{{F}^{21}}\frac{1}{c}{{v}^{2}}-{{F}^{13}}\frac{1}{c}{{v}^{3}} \right) \\ | ||
& =\frac{q}{\gamma }\left( {{F}^{10}}\gamma +{{F}^{21}}\frac{\gamma }{c}{{v}^{2}}-{{F}^{13}}\frac{\gamma }{c}{{v}^{3}} \right)=\frac{q}{\gamma }{{F}^{1\mu }}{{u}_{\mu }} \\ | & =\frac{q}{\gamma }\left( {{F}^{10}}\gamma +{{F}^{21}}\frac{\gamma }{c}{{v}^{2}}-{{F}^{13}}\frac{\gamma }{c}{{v}^{3}} \right)=\frac{q}{\gamma }{{F}^{1\mu }}{{u}_{\mu }} \\ | ||
\end{align}</math> mit <math>ds=\frac{c}{\gamma }dt</math> | \end{align}</math> | ||
mit | |||
<math>ds=\frac{c}{\gamma }dt</math> | |||
: | : | ||
<math>\frac{d}{ds}{{p}^{1}}=\frac{q}{c}{{F}^{1\mu }}{{u}_{\mu }}</math> | |||
Die zeitartige Komponente | Die zeitartige Komponente | ||
<math>\mu =0</math> | |||
gibt wegen | gibt wegen | ||
<math>{{p}^{0}}=\frac{E}{c}</math> | |||
: | : | ||
<math>\begin{align} | |||
& \frac{d}{ds}\frac{E}{c}=\frac{\gamma }{{{c}^{2}}}\frac{dE}{dt}=\frac{q}{c}\left( {{F}^{01}}{{u}_{1}}+{{F}^{02}}{{u}_{2}}+{{F}^{03}}{{u}_{3}} \right)= \\ | & \frac{d}{ds}\frac{E}{c}=\frac{\gamma }{{{c}^{2}}}\frac{dE}{dt}=\frac{q}{c}\left( {{F}^{01}}{{u}_{1}}+{{F}^{02}}{{u}_{2}}+{{F}^{03}}{{u}_{3}} \right)= \\ | ||
& =\frac{q\gamma }{{{c}^{2}}}\left( -{{E}^{1}}{{v}_{1}}-{{E}^{2}}{{v}_{2}}-{{E}^{3}}{{v}_{3}} \right)=\frac{q\gamma }{{{c}^{2}}}\left( {{E}^{1}}{{v}^{1}}+{{E}^{2}}{{v}^{2}}+{{E}^{3}}{{v}^{3}} \right) \\ | & =\frac{q\gamma }{{{c}^{2}}}\left( -{{E}^{1}}{{v}_{1}}-{{E}^{2}}{{v}_{2}}-{{E}^{3}}{{v}_{3}} \right)=\frac{q\gamma }{{{c}^{2}}}\left( {{E}^{1}}{{v}^{1}}+{{E}^{2}}{{v}^{2}}+{{E}^{3}}{{v}^{3}} \right) \\ |