"Electromagnetics"의 두 판 사이의 차이

2013년 3월 23일 (토) 04:52 판

gauge invariance

the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
the electromagnetism is a gauge field theory with structure group U(1)

Lorentz force

almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
Lorentz force is a rare example of velocity dependent force

polarization of light

has two possibilites
- what does this mean?

Lagrangian formulation

Lagrangian for a charged particle in an electromagnetic field $L=T-V$

\[L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\]

action

\[S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\]

Euler-Lagrange equations

\[p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\] $$ F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j} $$

equation of motion
$\dot{p}=F$ Therefore we get

\[m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\]. This is what we call the Lorentz force law.

force on a particle is same as $e\mathbf{E}+e\mathbf{v}\times \mathbf{B}$

Hamiltonian formulation

total energy of a charge particle in an electromagnetic field
$E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi$
replace the momentum with the canonical momentum
- similar to covariant derivative

force on a particle

force on a particle is same as $e\mathbf{E}+e\mathbf{v}\times \mathbf{B}$

메모

http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects

related items

encyclopedia

books

ELECTROMAGNETIC THEORY AND COMPUTATION

@@ 30번째 줄: / 30번째 줄: @@
 ==Lagrangian formulation==
-*  Lagrangian for a charged particle in an electromagnetic field<br><math>L=T-V</math><br><math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br>
+*  Lagrangian for a charged particle in an electromagnetic field <math>L=T-V</math>
-*  action<br><math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math><br>
+:<math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br>
-*  Euler-Lagrange equations<br><math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}</math><br><math>F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}</math><br>
+*  action
-*  equation of motion<br><math>\dot{p}=F</math> Therefore we get<br><math>m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}</math>. This is what we call the Lorentz force law.<br>
+:<math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math><br>
+*  Euler-Lagrange equations
+:<math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}</math>
+$$
+F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}
+$$
+*  equation of motion<br><math>\dot{p}=F</math> Therefore we get
+:<math>m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}</math>. This is what we call the Lorentz force law.<br>
 * force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math>