"Electromagnetics"의 두 판 사이의 차이
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Lagrangian formulation==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==gauge invariance | + | ==gauge invariance== |
* the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br> | * the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br> | ||
| 8번째 줄: | 8번째 줄: | ||
| − | ==Lorentz force | + | ==Lorentz force== |
* almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities | * almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities | ||
| 17번째 줄: | 17번째 줄: | ||
| − | ==polarization of light | + | ==polarization of light== |
* has two possibilites<br> | * has two possibilites<br> | ||
| 28번째 줄: | 28번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">Lagrangian formulation | + | <h5 style="margin: 0px; line-height: 2em;">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<br><math>L=T-V</math><br><math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br> | ||
| 43번째 줄: | 43번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">Hamiltonian formulation | + | <h5 style="margin: 0px; line-height: 2em;">Hamiltonian formulation== |
* total energy of a charge particle in an electromagnetic field<br><math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math><br> | * total energy of a charge particle in an electromagnetic field<br><math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math><br> | ||
| 53번째 줄: | 53번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">force on a particle | + | <h5 style="margin: 0px; line-height: 2em;">force on a particle== |
* force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | * force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | ||
| 65번째 줄: | 65번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">메모 | + | <h5 style="margin: 0px; line-height: 2em;">메모== |
* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]<br> | * [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]<br> | ||
| 74번째 줄: | 74번째 줄: | ||
| − | ==related items | + | ==related items== |
* [[Gauge theory]]<br> | * [[Gauge theory]]<br> | ||
| 83번째 줄: | 83번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 97번째 줄: | 97번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">books | + | <h5 style="margin: 0px; line-height: 2em;">books== |
ELECTROMAGNETIC THEORY AND COMPUTATION | ELECTROMAGNETIC THEORY AND COMPUTATION | ||
2012년 10월 28일 (일) 15:26 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==
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Classical_electromagnetism
- http://en.wikipedia.org/wiki/Maxwell's_equations
- http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations
- http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
- http://en.wikipedia.org/wiki/electrical_current
- http://en.wikipedia.org/wiki/Four-current
books==
ELECTROMAGNETIC THEORY AND COMPUTATION
\(L=T-V\)
\(L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\)
\(S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\)
\(p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\)
\(F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}\)
\(\dot{p}=F\) Therefore we get
\(m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\). This is what we call the Lorentz force law.
- 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
- similar to covariant derivative
- force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
- 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
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Classical_electromagnetism
- http://en.wikipedia.org/wiki/Maxwell's_equations
- http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations
- http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
- http://en.wikipedia.org/wiki/electrical_current
- http://en.wikipedia.org/wiki/Four-current