Editing Relativistische Formulierung der Elektrodynamik
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</noinclude>{{Scripthinweis|Elektrodynamik|6|0}}</noinclude> | {{Scripthinweis|Elektrodynamik|6}} | ||
= Ko- und Kontravariante Schreibweise der Relativitätstheorie= | |||
Grundpostulat der speziellen Relativitätstheorie: | |||
Kein Inertialsystem ist gegenüber einem anderen ausgezeichnet ! ( Einstein, 1904). | |||
Die Lichtgeschwindigkeit c ist in jedem Inertialsystem gleich ! | |||
* Kugelwellen sind | |||
* -> Lorentz- Invariant, also: | |||
* | |||
* <math>{{r}^{2}}-{{c}^{2}}{{t}^{2}}=r{{\acute{\ }}^{2}}-{{c}^{2}}t{{\acute{\ }}^{2}}</math> | |||
* | |||
Für Lorentz- Transformationen ! | |||
<u>'''Formalisierung:'''</u> | |||
<u>'''Der Raumzeitliche Abstand als'''</u> | |||
<math>{{\left( ds \right)}^{2}}:={{\left( cdt \right)}^{2}}-{{\left( d\bar{r} \right)}^{2}}</math> | |||
Zwischen 2 Ereignissen bleibt der raumzeitliche Abstand invariant bei Lorentz- Transformationen ! zwischen den Inertialsystemen : | |||
<math>\Sigma \leftrightarrow \Sigma \acute{\ }</math> | |||
Ziel: Um dies sofort zu sehen führt man Vierervektoren ein. | |||
Dann schreibt man | |||
<math>{{\left( ds \right)}^{2}}</math> | |||
als Skalarprodukt von Vierervektoren im Minkowski- Raum V und man benutze den Formalismus der '''linearen orthogonalen '''Transformation , unter denen das Skalarprodukt invariant bleibt: | |||
In der ko / kontravarianten Schreibweise tritt jeder Vierervektor in 2 möglichen Versionen auf: | |||
<u>'''kontravariante Komponenten:'''</u> | |||
<math>\begin{align} | |||
& {{x}^{i}} \\ | |||
& {{x}^{1}}:=ct \\ | |||
& {{x}^{1}},{{x}^{2}},{{x}^{3}} \\ | |||
\end{align}</math> | |||
als Komponenten des Ortsvektors | |||
<math>\bar{r}</math> | |||
: | |||
<u>'''kovariante Komponenten'''</u> | |||
<math>\begin{align} | |||
& {{x}_{i}}: \\ | |||
& {{x}_{0}}:=ct \\ | |||
& {{x}_{\alpha }}=-{{x}^{\alpha }},\alpha =1,2,3 \\ | |||
\end{align}</math> | |||
kovarianter Vektor | |||
<math>\in \tilde{V}</math> | |||
, dualer Vektorraum zu V ! | |||
Merke: Die Räume der kovarianten Vektoren ist dual zur menge der kontravarianten | |||
-> | |||
<math>\in \tilde{V}</math> | |||
als Raum der linearen Funktionale l: | |||
<math>V\to R</math> | |||
Damit werden dann die Skalarprodukte gebildet ! | |||
Schreibe | |||
<math>{{\left( ds \right)}^{2}}=d{{x}^{0}}d{{x}_{0}}+d{{x}^{1}}d{{x}_{1}}+d{{x}^{2}}d{{x}_{2}}+d{{x}^{3}}d{{x}_{3}}=d{{x}^{i}}d{{x}_{i}}</math> | |||
Mit: Summenkonvention ! | |||
über je einen ko- und einen kontravarianten Index ( hier i =0,1,2,3) wird summiert ! | |||
<u>'''Physikalische Anwendung'''</u> | |||
Lorentz- Invarianten lassen sich als Skalarprodukt | |||
<math>{{a}^{i}}{{a}_{i}}</math> | |||
schreiben ! | |||
'''Beispiel: dÁlemebert- Operator:''' | |||
<math>\#=\Delta -\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}=-\frac{\partial }{\partial {{x}^{i}}}\frac{\partial }{\partial {{x}_{i}}}=-{{\partial }_{i}}{{\partial }^{i}}</math> | |||
<u>'''Vierergeschwindigkeit'''</u> | |||
<math>\begin{align} | |||
& {{u}^{i}}:=\frac{d{{x}^{i}}}{ds}\Rightarrow {{u}^{i}}{{u}_{i}}=\frac{d{{x}^{i}}d{{x}_{i}}}{{{\left( ds \right)}^{2}}}=1 \\ | |||
& mit \\ | |||
& ds={{\left( d{{x}^{i}}d{{x}_{i}} \right)}^{\frac{1}{2}}}=c{{\left( 1-{{\beta }^{2}} \right)}^{\frac{1}{2}dt}}=\frac{c}{\gamma }dt \\ | |||
& \Rightarrow {{u}^{0}}=\gamma \\ | |||
& {{u}^{\alpha }}=\frac{\gamma }{c}{{v}^{\alpha }} \\ | |||
& {{v}^{\alpha }}:=\frac{d{{x}^{\alpha }}}{dt} \\ | |||
& \beta :=\frac{v}{c} \\ | |||
& \gamma :=\frac{1}{\sqrt{1-{{\beta }^{2}}}} \\ | |||
\end{align}</math> | |||
'''Physikalische Interpretation''' | |||
<math>\begin{align} | |||
& {{u}^{\alpha }}=\frac{1}{c}\frac{d{{x}^{\alpha }}}{d\tau } \\ | |||
& d\tau =\frac{dt}{\gamma } \\ | |||
\end{align}</math> | |||
'''Viererimpuls''' | |||
<math>{{p}^{i}}:={{m}_{0}}c{{u}^{i}}</math> | |||
mit der Ruhemasse m0 | |||
Also: | |||
<math>\begin{align} | |||
& {{p}^{i}}{{p}_{i}}={{m}_{0}}^{2}{{c}^{2}}{{u}^{i}}{{u}_{i}} \\ | |||
& {{u}^{i}}{{u}_{i}}=1 \\ | |||
& \Rightarrow {{p}^{i}}{{p}_{i}}={{m}_{0}}^{2}{{c}^{2}} \\ | |||
& {{p}^{0}}={{m}_{0}}\gamma c=m(v)c=\frac{E}{c} \\ | |||
& {{p}^{\alpha }}={{m}_{0}}\gamma {{v}^{\alpha }}=m(v){{v}^{\alpha }} \\ | |||
& {{p}^{i}}{{p}_{i}}={{m}_{0}}^{2}{{c}^{2}}{{u}^{i}}{{u}_{i}}\Leftrightarrow {{E}^{2}}={{m}_{0}}^{2}{{c}^{4}}+{{c}^{2}}{{{\bar{p}}}^{2}} \\ | |||
\end{align}</math> | |||
Mit der Energie | |||
<math>E=m(v){{c}^{2}}</math> | |||
'''Analoge Definition von Tensoren 2. Stufe:''' | |||
<math>\begin{align} | |||
& {{A}^{ik}},{{A}^{i}}_{k},{{A}_{i}}^{k},{{A}_{ik}} \\ | |||
& {{A}^{00}}={{A}^{0}}_{0}={{A}_{0}}^{0}={{A}_{00}} \\ | |||
& {{A}^{10}}={{A}^{1}}_{0}=-{{A}_{1}}^{0}=-{{A}_{10}} \\ | |||
& {{A}^{11}}=-{{A}^{1}}_{1}=-{{A}_{1}}^{1}={{A}_{11}} \\ | |||
\end{align}</math> | |||
<u>'''Der metrische Tensor'''</u> | |||
<math>{{g}^{ik}}:={{\delta }^{ik}}=\left. \left\{ \begin{matrix} | |||
{{\delta }^{i}}_{k}\quad k=0 \\ | |||
-{{\delta }^{i}}_{k}\quad k=1,2,3 \\ | |||
\end{matrix} \right. \right\}={{g}_{ik}}</math> | |||
<math>{{g}^{ik}}={{g}_{ik}}=\left( \begin{matrix} | |||
1 & 0 & 0 & 0 \\ | |||
0 & -1 & 0 & 0 \\ | |||
0 & 0 & -1 & 0 \\ | |||
0 & 0 & 0 & -1 \\ | |||
\end{matrix} \right)</math> | |||
Mittels der Metrik werden Indices gehoben bzw. gesenkt: | |||
<math>{{g}^{ik}}{{a}_{k}}={{a}^{i}}</math> | |||
Wichtig fürs Skalarprodukt: | |||
<math>d{{s}^{2}}={{g}^{ik}}d{{x}_{i}}d{{x}_{k}}={{g}_{ik}}d{{x}^{i}}d{{x}^{k}}</math> | |||
<u>Lorentz- Trafo</u> | |||
zwischen Bezugssystemen: Lineare / homogene Trafo | |||
die Lorentz- Transformation für | |||
<math>\begin{align} | |||
& \left( {{x}^{0}}\begin{matrix} | |||
, & {{x}^{1}}, & {{x}^{2}}, & {{x}^{3}} \\ | |||
\end{matrix} \right)=\left( \begin{matrix} | |||
ct, & x, & y, & z \\ | |||
\end{matrix} \right) \\ | |||
& d{{s}^{2}}={{c}^{2}}d{{t}^{2}}-d{{x}^{2}}-d{{y}^{2}}-d{{z}^{2}} \\ | |||
\end{align}</math> | |||
Nämlich: | |||
<math>\begin{align} | |||
& \left( \begin{matrix} | |||
{{x}_{0}}\acute{\ } \\ | |||
{{x}_{1}}\acute{\ } \\ | |||
{{x}_{2}}\acute{\ } \\ | |||
{{x}_{3}}\acute{\ } \\ | |||
\end{matrix} \right)=\left( \begin{matrix} | |||
\frac{1}{\sqrt{1-{{\beta }^{2}}}} & \frac{-\beta }{\sqrt{1-{{\beta }^{2}}}} & 0 & 0 \\ | |||
\frac{-\beta }{\sqrt{1-{{\beta }^{2}}}} & \frac{1}{\sqrt{1-{{\beta }^{2}}}} & 0 & 0 \\ | |||
0 & 0 & 1 & 0 \\ | |||
0 & 0 & 0 & 1 \\ | |||
\end{matrix} \right)\left( \begin{matrix} | |||
{{x}_{0}} \\ | |||
{{x}_{1}} \\ | |||
{{x}_{2}} \\ | |||
{{x}_{3}} \\ | |||
\end{matrix} \right) \\ | |||
& x{{\acute{\ }}^{i}}={{U}^{i}}_{k}{{x}^{k}} \\ | |||
\end{align}</math> | |||
Mit | |||
<math>{{U}^{i}}_{k}=\left( \begin{matrix} | |||
\frac{1}{\sqrt{1-{{\beta }^{2}}}} & \frac{-\beta }{\sqrt{1-{{\beta }^{2}}}} & 0 & 0 \\ | |||
\frac{-\beta }{\sqrt{1-{{\beta }^{2}}}} & \frac{1}{\sqrt{1-{{\beta }^{2}}}} & 0 & 0 \\ | |||
0 & 0 & 1 & 0 \\ | |||
0 & 0 & 0 & 1 \\ | |||
\end{matrix} \right)</math> | |||
für | |||
<math>v||{{x}_{1}}</math> | |||
Wesentliche Eigenschaft ( die Viererschreibweise ist so konstruiert worden): | |||
U ist orthogonale Trafo: | |||
<math>\begin{align} | |||
& {{U}^{i}}_{k}{{U}_{i}}^{l}=\delta _{k}^{l} \\ | |||
& \Rightarrow a{{\acute{\ }}^{i}}b{{\acute{\ }}_{i}}={{U}^{i}}_{k}{{U}_{i}}^{l}{{a}^{k}}{{b}_{l}}={{a}^{k}}{{b}_{k}} \\ | |||
\end{align}</math> | |||
Das Skalarprodukt ist invariant, falls U eine orthogonale Trafo ist | |||
Bzw. | |||
Forderung: Skalarprodukt invariant -> U muss orthogonale Trafo sein ! | |||
Umkehr- Transformation: | |||
<math>{{x}^{i}}={{U}_{k}}^{i}x{{\acute{\ }}^{k}}</math> | |||
<noinclude>{{Scripthinweis|Elektrodynamik|6|0}}</noinclude> | |||
=Inhomogene Maxwellgleichungen im Vakuum= | |||
( Erregungsgleichungen) | |||
<math>\begin{align} | |||
& {{\varepsilon }_{0}}\nabla \cdot \bar{E}=\rho \\ | |||
& \Leftrightarrow {{\partial }_{1}}{{E}^{1}}+{{\partial }_{2}}{{E}^{2}}+{{\partial }_{3}}{{E}^{3}}=\frac{1}{{{\varepsilon }_{0}}c}c\rho \\ | |||
& \Leftrightarrow {{\partial }_{1}}{{F}^{10}}+{{\partial }_{2}}{{F}^{20}}+{{\partial }_{3}}{{F}^{30}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{0}} \\ | |||
& \Leftrightarrow {{\partial }_{\nu }}{{F}^{\nu 0}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{0}} \\ | |||
& wegen{{\partial }_{0}}{{F}^{00}}=0 \\ | |||
& auch{{\partial }_{i}}{{F}^{i0}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{0}} \\ | |||
\end{align}</math> | |||
# | |||
# <math>\nabla \times \bar{B}-\frac{1}{{{c}^{2}}}\frac{\partial }{\partial t}\bar{E}={{\mu }_{0}}\left( \nabla \times \bar{H}-{{\varepsilon }_{0}}\frac{\partial }{\partial t}\bar{E} \right)={{\mu }_{0}}\bar{j}</math> | |||
# | |||
# Komponente | |||
<math>\begin{align} | |||
& {{\partial }_{2}}{{B}^{3}}-{{\partial }_{3}}{{B}^{2}}={{\mu }_{0}}{{j}^{1}}+{{\varepsilon }_{0}}{{\mu }_{0}}\frac{\partial }{\partial t}{{E}^{1}} \\ | |||
& {{\mu }_{0}}c=\frac{1}{{{\varepsilon }_{0}}c} \\ | |||
& \Leftrightarrow {{\partial }_{2}}{{F}^{21}}-.{{\partial }_{3}}{{F}^{13}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{1}}+.{{\partial }_{0}}{{F}^{10}} \\ | |||
& {{\partial }_{2}}{{F}^{21}}+{{\partial }_{3}}{{F}^{31}}+{{\partial }_{0}}{{F}^{01}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{1}} \\ | |||
& \Leftrightarrow {{\partial }_{\nu }}{{F}^{\nu 1}}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{1}} \\ | |||
& wegen{{\partial }_{1}}{{F}^{11}}=0 \\ | |||
\end{align}</math> | |||
Dies kann analog für die zweite und dritte Komponente durchgeixt werden. Aus der Nullten Komponente hatten wir die Nullte des Stroms ( Erregungsgleichung des elektrischen Feldes), so dass insgesamt folgt: | |||
<math>\begin{align} | |||
& {{\partial }_{\nu }}{{F}^{\mu \nu }}=-\frac{1}{{{\varepsilon }_{0}}c}{{j}^{\mu }} \\ | |||
& {{\partial }_{\nu }}{{F}^{\nu \mu }}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{\mu }} \\ | |||
\end{align}</math> | |||
Die Viererdivergenz des elektrischen Feldstärketensors ! | |||
'''Bemerkungen''' | |||
# die homogenen Maxwellgleichungen sind durch den Potenzialansatz | |||
<math>\left\{ {{F}_{\mu \nu }} \right\}=\left\{ {{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\partial }_{\nu }}{{\Phi }_{\mu }} \right\}=\left( \begin{matrix} | |||
0 & \frac{1}{c}{{E}_{x}} & \frac{1}{c}{{E}_{y}} & \frac{1}{c}{{E}_{z}} \\ | |||
-\frac{1}{c}{{E}_{x}} & 0 & -{{B}_{z}} & {{B}_{y}} \\ | |||
-\frac{1}{c}{{E}_{y}} & {{B}_{z}} & 0 & -{{B}_{x}} \\ | |||
-\frac{1}{c}{{E}_{z}} & -{{B}_{y}} & {{B}_{x}} & 0 \\ | |||
\end{matrix} \right)</math> | |||
automatisch erfüllt: | |||
<math>\begin{align} | |||
& {{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{F}_{\mu \nu }}={{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\nu }}{{\Phi }_{\mu }} \\ | |||
& {{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\mu }}{{\Phi }_{\nu }}=0, \\ | |||
& da:{{\partial }_{\beta }}{{\partial }_{\mu }}{{\Phi }_{\nu }}\quad symmetrisch \\ | |||
& {{\varepsilon }^{\alpha \beta \mu \nu }}\quad antisymmetrisch \\ | |||
& {{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\nu }}{{\Phi }_{\mu }}=0 \\ | |||
\end{align}</math> | |||
Aus den inhomogenen Maxwell- Gleichungen | |||
<math>{{\partial }_{\beta }}{{F}^{\beta \nu }}={{\partial }_{\beta }}{{\partial }^{\beta }}{{\Phi }^{\nu }}-{{\partial }_{\beta }}{{\partial }^{\nu }}{{\Phi }^{\beta }}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{\nu }}</math> | |||
folgt mit Lorentz- Eichung | |||
<math>{{\partial }_{\mu }}{{\Phi }^{\mu }}=0</math> | |||
<math>\begin{align} | |||
& {{\partial }_{\beta }}{{\partial }^{\nu }}{{\Phi }^{\beta }}={{\partial }^{\nu }}{{\partial }_{\beta }}{{\Phi }^{\beta }}=0 \\ | |||
& also: \\ | |||
\end{align}</math> | |||
<math>{{\partial }_{\beta }}{{F}^{\beta \nu }}={{\partial }_{\beta }}{{\partial }^{\beta }}{{\Phi }^{\nu }}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{\nu }}</math> | |||
als inhomogene Wellengleichung | |||
'''Die Maxwellgleichungen''' | |||
<math>\begin{align} | |||
& {{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{F}_{\mu \nu }}={{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\varepsilon }^{\alpha \beta \mu \nu }}{{\partial }_{\beta }}{{\partial }_{\nu }}{{\Phi }_{\mu }}=0 \\ | |||
& {{\partial }_{\beta }}{{F}^{\beta \nu }}={{\partial }_{\beta }}{{\partial }^{\beta }}{{\Phi }^{\nu }}=\frac{1}{{{\varepsilon }_{0}}c}{{j}^{\nu }} \\ | |||
\end{align}</math> | |||
sind ihrerseits nun Lorentz- kovariant, da sie durch 4 Pseudovektoren ausgedrückt sind. | |||
Merke: Pseudo - 4- Vektor stört nicht, da rechte Seite gleich Null !! | |||
<u>'''Gauß- System:'''</u> | |||
<math>{{\partial }_{\beta }}{{F}^{\beta \nu }}=\frac{4\pi }{c}{{j}^{\nu }}</math> | |||
=Relativistisches Hamiltonprinzip= | |||
<u>'''Ziel: '''</u>Formulierung der Elektrodynamik als Lagrange- Feldtheorie | |||
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 \\ | |||
& W=\int_{1}^{2}{{}}ds \\ | |||
\end{align}</math> | |||
letzteres: Wirkungsintegral | |||
Wichtig: | |||
<math>{{\left. \delta {{x}^{i}} \right|}_{1,2}}=0</math> | |||
Newtonsche Mechanik ist Grenzfall: | |||
<math>W=-{{m}_{0}}c\int_{1}^{2}{{}}ds</math> | |||
Wechselwirkung eines Massepunktes mit einem 4- Vektor- Feld | |||
<math>\begin{align} | |||
& \left( {{\phi }^{i}} \right)({{x}^{j}}) \\ | |||
& \Rightarrow \\ | |||
\end{align}</math> | |||
<math>W=\int_{1}^{2}{{}}\left\{ -{{m}_{0}}cds-{{\phi }^{i}}d{{x}_{i}} \right\}</math> | |||
mit den Lorentz- Invarianten | |||
<math>{{m}_{0}}cds</math> | |||
und | |||
<math>{{\phi }^{i}}d{{x}_{i}}</math> | |||
'''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: | |||
<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} \\ | |||
& \left( d\delta {{x}^{\mu }} \right)d{{x}_{\mu }}=d{{x}^{\mu }}\left( d\delta {{x}_{\mu }} \right) \\ | |||
& =\frac{d{{x}^{\mu }}}{ds}\left( d\delta {{x}_{\mu }} \right)={{u}^{\mu }}\left( d\delta {{x}_{\mu }} \right) \\ | |||
\end{align}</math> | |||
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: | |||
<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: | |||
<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) \\ | |||
& \left[ -{{m}_{0}}c{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right) \right]_{1}^{2}=0,weil\delta {{x}_{\mu }}_{1}^{2}=0 \\ | |||
& \Rightarrow \int_{1}^{2}{{}}-{{m}_{0}}c{{u}^{\mu }}d\left( \delta {{x}_{\mu }} \right)=\int_{1}^{2}{{}}{{m}_{0}}cd{{u}^{\mu }}\left( \delta {{x}_{\mu }} \right)=\int_{1}^{2}{{}}{{m}_{0}}c\frac{d{{u}^{\mu }}}{ds}\left( \delta {{x}_{\mu }} \right)ds \\ | |||
\end{align}</math> | |||
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 | |||
<math>\begin{align} | |||
& 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 }}d{{x}_{\mu }}={{\partial }^{\nu }}{{\phi }^{\mu }}\delta {{x}_{\nu }}d{{x}_{\mu }}=i<->k={{\partial }^{\mu }}{{\phi }^{\nu }}\delta {{x}_{\mu }}d{{x}_{\nu }}={{\partial }^{\mu }}{{\phi }^{\nu }}{{u}_{\nu }}\delta {{x}_{\mu }}ds \\ | |||
\end{align}</math> | |||
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: | |||
<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''' | |||
<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 \\ | |||
& {{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 }} \\ | |||
& {{f}^{\mu \nu }}=\left( {{\partial }^{\mu }}{{\phi }^{\nu }}-{{\partial }^{\nu }}{{\phi }^{\mu }} \right) \\ | |||
\end{align}</math> | |||
Dies ist dann die aus dem hamiltonschen Prinzip abgeleitete Bewegungsgleichung eines Massepunktes der Ruhemasse m0 und der Ladung q unter dem Einfluss der Lorentz- Kraft. | |||
Man setze: | |||
<math>\begin{align} | |||
& {{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) \\ | |||
& {{\phi }^{\mu }}=\frac{q}{c}{{\Phi }^{\mu }} \\ | |||
& \frac{d}{ds}{{p}^{\mu }}=\frac{q}{c}{{F}^{\mu \nu }}{{u}_{\nu }}\Leftrightarrow \delta W=\delta \int_{1}^{2}{{}}\left\{ -{{m}_{0}}cds-\frac{q}{c}{{\Phi }^{\mu }}d{{x}_{\mu }} \right\}=0 \\ | |||
\end{align}</math> | |||
Man bestimmt die Ortskomponenten | |||
<math>\alpha =1,2,3</math> | |||
über | |||
<math>\begin{align} | |||
& \frac{d}{dt}\bar{p}=q\left( \bar{E}+\bar{v}\times \bar{B} \right) \\ | |||
& \\ | |||
\end{align}</math> | |||
überein, denn mit | |||
<math>\begin{align} | |||
& {{u}^{0}}=\gamma \\ | |||
& {{u}^{\alpha }}=\frac{\gamma }{c}{{v}^{\alpha }}=-{{u}_{\alpha }} \\ | |||
\end{align}</math> | |||
folgt dann: | |||
<math>\begin{align} | |||
& \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) \\ | |||
& =\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> | |||
: | |||
<math>\frac{d}{ds}{{p}^{1}}=\frac{q}{c}{{F}^{1\mu }}{{u}_{\mu }}</math> | |||
Die zeitartige Komponente | |||
<math>\mu =0</math> | |||
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{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{dE}{dt}=q\bar{E}\cdot \bar{v} \\ | |||
\end{align}</math> | |||
Dies ist die Leistungsbilanz: Die Änderung der inneren Energie ist gleich der reingesteckten Arbeit | |||
=Eichinvarianz und Ladungserhaltung= | |||
Wirkungsintegral: | |||
<math>W=-{{m}_{0}}c\int_{1}^{2}{{}}ds-\frac{q}{c}\int_{1}^{2}{{}}d{{x}_{\mu }}{{\Phi }^{\mu }}</math> | |||
Dabei: | |||
<math>{{m}_{0}}c\int_{1}^{2}{{}}ds={{W}_{t}}</math> | |||
( Teilchen) | |||
<math>-\frac{q}{c}\int_{1}^{2}{{}}d{{x}_{\mu }}{{\Phi }^{\mu }}={{W}_{tf}}</math> | |||
( Teilchen- Feld- Wechselwirkung) | |||
Verallgemeinerung auf kontinuierliche Massendichte | |||
<math>m\left( {{x}^{\mu }} \right)</math> | |||
: | |||
Vorsicht: m ist hier Massendichte !!! | |||
<math>\begin{align} | |||
& {{W}_{t}}=-c\int_{{}}^{{}}{{}}{{d}^{3}}rm\int_{1}^{2}{{}}ds=-\int_{\Omega }^{{}}{{}}d\Omega m\frac{ds}{dt} \\ | |||
& d\Omega :={{d}^{3}}rcdt=d{{x}^{0}}d{{x}^{1}}d{{x}^{2}}d{{x}^{3}} \\ | |||
\end{align}</math> | |||
dOmega als Volumenelement im Minkowski- Raum !!! | |||
Bemerkungen: | |||
# | |||
# <math>d\Omega </math> | |||
# ist eine Lorentz- Invariante , da das Volumen unter orthogonalen Transformationen | |||
<math>{{U}^{\mu }}_{\nu }</math> | |||
erhalten bleibt. | |||
2) Aus | |||
<math>d{{m}_{0}}d{{x}^{\mu }}=\frac{\mu }{c}\frac{d{{x}^{\mu }}}{dt}{{d}^{3}}rcdt;{{d}^{3}}rcdt=d\Omega \Rightarrow d{{m}_{0}}d{{x}^{\mu }}=\frac{\mu }{c}\frac{d{{x}^{\mu }}}{dt}d\Omega </math> | |||
folgt, dass die Vierer- Massenstromdichte mit Massendichte m= | |||
<math>d{{m}_{0}}d{{x}^{\mu }}=\frac{\mu }{c}\frac{d{{x}^{\mu }}}{dt}{{d}^{3}}rcdt;{{d}^{3}}rcdt=d\Omega \Rightarrow d{{m}_{0}}d{{x}^{\mu }}=\frac{\mu }{c}\frac{d{{x}^{\mu }}}{dt}d\Omega </math> | |||
: | |||
<math>{{m}_{0}}\frac{d{{x}^{\mu }}}{dt}\equiv {{g}^{\mu }}</math> | |||
ein Vier- Vektor ist, da | |||
<math>d{{m}_{0}},d\Omega </math> | |||
Lorentz- Skalare sind und natürlich | |||
<math>d{{x}^{\mu }}</math> | |||
selbst auch ein Vierervektor | |||
# | |||
# <math>{{\mu }^{2}}\frac{d{{x}^{\mu }}d{{x}_{\mu }}}{{{\left( dt \right)}^{2}}}={{g}^{\mu }}{{g}_{\mu }}={{\left( \mu \frac{ds}{dt} \right)}^{2}}</math> | |||
# ist Lorentz - Invariant. | |||
Also | |||
<math>{{g}^{\mu }}{{g}_{\mu }}</math> | |||
ist Lorentz- Invariant. Also auch | |||
<math>\left( \mu \frac{ds}{dt} \right)</math> | |||
. | |||
Somit ist | |||
<math>{{W}_{t}}</math> | |||
insgesamt Lorentz- Invariant ! |