"Classical field theory and classical mechanics"의 두 판 사이의 차이
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notation==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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1번째 줄: | 1번째 줄: | ||
− | ==introduction | + | ==introduction== |
* can be formulated using classical fields and Lagrangian density | * can be formulated using classical fields and Lagrangian density | ||
14번째 줄: | 14번째 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%;">notation | + | <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%;">notation== |
* dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math><br> | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math><br> | ||
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− | ==Lagrangian formalism | + | ==Lagrangian formalism== |
* [[Lagrangian formalism]] | * [[Lagrangian formalism]] | ||
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− | <h5 style="margin: 0px; line-height: 2em;">canonically conjugate momentum | + | <h5 style="margin: 0px; line-height: 2em;">canonically conjugate momentum== |
* canonically conjugate momenta<br><math>p_{k}=\frac{\partial L}{\partial \dot{q}_k}</math><br> | * canonically conjugate momenta<br><math>p_{k}=\frac{\partial L}{\partial \dot{q}_k}</math><br> | ||
48번째 줄: | 48번째 줄: | ||
− | <h5 style="margin: 0px; line-height: 2em;">Hamiltonian mechanics | + | <h5 style="margin: 0px; line-height: 2em;">Hamiltonian mechanics== |
* conjugate variables are on the equal footing<br> | * conjugate variables are on the equal footing<br> | ||
59번째 줄: | 59번째 줄: | ||
− | <h5 style="margin: 0px; line-height: 2em;">Poisson bracket | + | <h5 style="margin: 0px; line-height: 2em;">Poisson bracket== |
For <math>f(p_i,q_i,t), g(p_i,q_i,t)</math> , we define the Poisson bracket | For <math>f(p_i,q_i,t), g(p_i,q_i,t)</math> , we define the Poisson bracket | ||
73번째 줄: | 73번째 줄: | ||
− | <h5 style="margin: 0px; line-height: 2em;">phase space | + | <h5 style="margin: 0px; line-height: 2em;">phase space== |
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− | ==links and webpages | + | ==links and webpages== |
* [http://www.astro.caltech.edu/%7Egolwala/ph106ab/ph106ab_notes.pdf ][http://www.astro.caltech.edu/%7Egolwala/ph106ab/ph106ab_notes.pdf http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf] | * [http://www.astro.caltech.edu/%7Egolwala/ph106ab/ph106ab_notes.pdf ][http://www.astro.caltech.edu/%7Egolwala/ph106ab/ph106ab_notes.pdf http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf] | ||
109번째 줄: | 109번째 줄: | ||
− | ==question and answers(Math Overflow) | + | ==question and answers(Math Overflow)== |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
121번째 줄: | 121번째 줄: | ||
− | ==history | + | ==history== |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
129번째 줄: | 129번째 줄: | ||
− | ==related items | + | ==related items== |
* [[Electromagnetics|Electromagnetism]] | * [[Electromagnetics|Electromagnetism]] | ||
140번째 줄: | 140번째 줄: | ||
− | ==encyclopedia | + | ==encyclopedia== |
* http://en.wikipedia.org/wiki/Classical_field_theory | * http://en.wikipedia.org/wiki/Classical_field_theory | ||
162번째 줄: | 162번째 줄: | ||
− | ==books | + | ==books== |
* <br> | * <br> | ||
173번째 줄: | 173번째 줄: | ||
− | ==expositions | + | ==expositions== |
* Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences | * Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences |
2012년 10월 28일 (일) 15:25 판
introduction
- can be formulated using classical fields and Lagrangian density
- change the coordinates and fields accordingly
- require the invariance of action integral over arbitrary region
- this invariance consists of two parts : Euler-Lagrange equation and the equation of continuity
- three important conserved quantity
- energy
- momentum
- angular momentum
notation==
- dynamical variables \(q_{k}, \dot{q}_k\) for \(k=1,\cdots, N\)
- \(T\) kinetic energy
- \(V\) potential energy
- We have Lagrangian \(L=T-V\)
- Define the Hamiltonian
- \(H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L\)
- \(p\dot q\) is twice of kinetic energy
- Thus the Hamiltonian represents \(H=T+V\) the total energy of the system
Lagrangian formalism
canonically conjugate momentum==
- canonically conjugate momenta
\(p_{k}=\frac{\partial L}{\partial \dot{q}_k}\)
- instead of \(q_{k}, \dot{q}_k\), one can use \(q_{k}, p_{k}\) as dynamical variables
Hamiltonian mechanics==
- conjugate variables are on the equal footing
- 고전역학에서의 가적분성 항목 참조
Poisson bracket==
For \(f(p_i,q_i,t), g(p_i,q_i,t)\) , we define the Poisson bracket
\(\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]\)
In quantization we have correspondence
\(\{f,g\} = \frac{1}{i}[u,v]\)
phase space==
하위페이지
links and webpages
question and answers(Math Overflow)
history
- Electromagnetism
- Einstein field hequation
- sympletic geometry
- integrable Hamiltonian systems and solvable models
encyclopedia
- http://en.wikipedia.org/wiki/Classical_field_theory
- http://en.wikipedia.org/wiki/Continuity_equation
- http://en.wikipedia.org/wiki/current_density
- http://en.wikipedia.org/wiki/Noether's_theorem
- http://ko.wikipedia.org /wiki/작용
- http://en.wikipedia.org/wiki/Canonical_coordinates
- http://en.wikipedia.org/wiki/Lagrangian_mechanics
- http://en.wikipedia.org/wiki/Lagrangian
- http://en.wikipedia.org/wiki/poisson_bracket
- http://en.wikipedia.org/wiki/Action_(physics)
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
-
- Classical mechanics Classical_Mechanics.djvuV.I. Arnold
- Emmy Noether’s Wonderful Theorem
-
Electrodynamics and Classical Theory of Fields and Particles
expositions
- Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences
- canonically conjugate momenta
\(p_{k}=\frac{\partial L}{\partial \dot{q}_k}\) - instead of \(q_{k}, \dot{q}_k\), one can use \(q_{k}, p_{k}\) as dynamical variables
Hamiltonian mechanics==
- conjugate variables are on the equal footing
- 고전역학에서의 가적분성 항목 참조
Poisson bracket== For \(f(p_i,q_i,t), g(p_i,q_i,t)\) , we define the Poisson bracket \(\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]\) In quantization we have correspondence \(\{f,g\} = \frac{1}{i}[u,v]\)
phase space==
하위페이지
links and webpages
question and answers(Math Overflow)
history
- Electromagnetism
- Einstein field hequation
- sympletic geometry
- integrable Hamiltonian systems and solvable models
encyclopedia
- http://en.wikipedia.org/wiki/Classical_field_theory
- http://en.wikipedia.org/wiki/Continuity_equation
- http://en.wikipedia.org/wiki/current_density
- http://en.wikipedia.org/wiki/Noether's_theorem
- http://ko.wikipedia.org /wiki/작용
- http://en.wikipedia.org/wiki/Canonical_coordinates
- http://en.wikipedia.org/wiki/Lagrangian_mechanics
- http://en.wikipedia.org/wiki/Lagrangian
- http://en.wikipedia.org/wiki/poisson_bracket
- http://en.wikipedia.org/wiki/Action_(physics)
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
-
- Classical mechanics Classical_Mechanics.djvuV.I. Arnold
- Emmy Noether’s Wonderful Theorem
-
Electrodynamics and Classical Theory of Fields and Particles
expositions
- Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences