"Lagrangian formulation of electromagetism"의 두 판 사이의 차이

2020년 11월 13일 (금) 10:43 기준 최신판

@@ 1번째 줄: / 1번째 줄: @@
-==하전입자에 대한 라그랑지안==
+* {{수학노트|url=전자기학의_라그랑지안}}
-* 전자기장 안에서 하전입자에 대한 라그랑지안
-:<math>L(q,\dot{q})=\frac{m||\dot{q}||^2}{2}-e\phi+eA_{i}\dot{q}^{i}</math>
-* 켤레운동량
-:<math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m \dot{q}_{i}+eA_{i}=mv_{i}+eA_{i}</math>
-$$
-\dot{p}_{i}=m\frac{dv_{i}}{dt}+e\frac{\partial{A_{i}}}{\partial t}+e\frac{\partial{A_{i}}}{\partial{q}^{j}}\dot{q}^{j}
-$$
-$$
-F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(-e\phi+eA_{j}\dot{q}^{j})=-e\frac{\partial{\phi}}{\partial{q}^{i}} +e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}
-$$
-* 오일러-라그랑지 방정식 <math>\dot{p}=F,</math>
-:<math>m\frac{dv_{i}}{dt}=eE_{i}+eF_{ij}\dot{q}^{j}</math>
-where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!$
-* for example, if $i=1$
-$$
-ma_1=eE_1+e(F_{11}\dot{q}^{1}+F_{12}\dot{q}^{2}+F_{13}\dot{q}^{3})=eE_1+e(F_{12}\dot{q}^{2}-F_{31}\dot{q}^{3})=eE_1+e(\mathbf{v}\times \mathbf{B})_{1}
-$$
-where $F_{12}=-B_{3}$ and $F_{31}=-B_{2}$
-* 이를 로렌츠 힘이라 한다
-* 전하가 받는 힘은
-:<math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math>
-* http://en.wikipedia.org/wiki/Lorentz_force
-* {{수학노트|url=전자기_텐서와_맥스웰_방정식}}
-==전자기장에 대한 라그랑지안==
-===free===
-* 상호작용이 없는 전자기장의 라그랑지안은 다음과 같다
-$$\mathcal{L}_{\text{EM}}= - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\mathbf{E}^2-\mathbf{B}^2)$$
-이 때 <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math>는 전자기텐서, $A=(A_{\mu})$는 전자기 포텐셜
-*  action
-:<math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math>
-* 라그랑지안은 전자기 포텐셜의 다음과 같은 변환에 대하여 불변이다
-:<math>A_{\mu}(x) \to A_{\mu}(x)-\partial_{\mu}\Lambda(x)</math>
-여기서 $\Lambda(x)$는 임의의 스칼라장
-* equation of motion
-$$
-\partial_\mu F^{\mu\nu}=0
-$$
-* see http://www.damtp.cam.ac.uk/user/tong/qft/six.pdf
-===in the presence of $j$ and $\rho$===
-* Lagrangian
-$$L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-ej_\mu A^\mu$$
-* action
-$$S[\phi,A]=\int_{t_1}^{t_2}\int_{\mathbb{R}^3}\left(-\rho\phi+j\cdot A+\frac{\epsilon_0}{2}E^2-\frac{1}{2\mu_0}B^2\right)\,dV\,dt$$
-* w.r.t $\phi$
-$$\nabla\cdot E=\frac{\rho}{\epsilon_0}$$
-* w.r.t $A$
-$$\nabla\times B=\mu_0j+\epsilon_0\mu_0\frac{\partial E}{\partial t}$$
-==memo==
-* http://dexterstory.tistory.com/888
-==related items==
 * [[path integral formulation of quantum mechanics]]
 * [[Electroweak theory]]
+[[분류:migrate]]
-==expositions==
-* THOMAS YU [http://math.uchicago.edu/~may/REU2012/REUPapers/Yu.pdf Lagrangian formulation of the electromagnetic field]
-* Lea, [http://www.physics.sfsu.edu/~lea/courses/grad/fldlagr.PDF The field Lagrangian]
-* Susskind, [http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/ The electromagnetic Lagrangian]
-* Lecture 8 | Modern Physics: Classical Mechanics (Stanford). 2008. http://www.youtube.com/watch?v=gUUbl444r74&feature=youtube_gdata_player.
-==blogs==
-* Higgs mechanism
-** http://unapologetic.wordpress.com/2012/07/16/the-higgs-mechanism-part-1-lagrangians/
-** http://unapologetic.wordpress.com/2012/07/17/the-higgs-mechanism-part-2-examples-of-lagrangian-field-equations/
-** http://unapologetic.wordpress.com/2012/07/18/the-higgs-mechanism-part-3-gauge-symmetries/
-** http://unapologetic.wordpress.com/2012/07/19/the-higgs-mechanism-part-4-symmetry-breaking/
-==questions==
-* http://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian?rq=1

"Lagrangian formulation of electromagetism"의 두 판 사이의 차이

2020년 11월 13일 (금) 10:43 기준 최신판

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