"Classical field theory and classical mechanics"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 5번째 줄: | 5번째 줄: | ||
* require the invariance of action integral over arbitrary region | * require the invariance of action integral over arbitrary region | ||
* this invariance consists of two parts : Euler-Lagrange equation and the equation of continuity | * this invariance consists of two parts : Euler-Lagrange equation and the equation of continuity | ||
| − | * three important conserved quantity | + | * three important conserved quantity |
** energy | ** energy | ||
** momentum | ** momentum | ||
** angular momentum | ** angular momentum | ||
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==notation== | ==notation== | ||
| − | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math | + | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math> |
| − | * <math>T</math> | + | * <math>T</math> kinetic energy |
| − | * <math>V</math> | + | * <math>V</math> potential energy |
| − | * We have | + | * We have Lagrangian <math>L=T-V</math> |
| − | * Define the Hamiltonian | + | * Define the Hamiltonian |
| − | * <math>H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L</math | + | * <math>H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L</math> |
| − | * <math>p\dot q</math> | + | * <math>p\dot q</math> is twice of kinetic energy |
| − | * Thus the Hamiltonian | + | * Thus the Hamiltonian represents <math>H=T+V</math> the total energy of the system |
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==Lagrangian formalism== | ==Lagrangian formalism== | ||
| 33번째 줄: | 33번째 줄: | ||
* [[Lagrangian formalism]] | * [[Lagrangian formalism]] | ||
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==canonically conjugate momentum== | ==canonically conjugate momentum== | ||
| − | * canonically conjugate momenta | + | * canonically conjugate momenta<math>p_{k}=\frac{\partial L}{\partial \dot{q}_k}</math> |
* instead of <math>q_{k}, \dot{q}_k</math>, one can use <math>q_{k}, p_{k}</math> as dynamical variables | * instead of <math>q_{k}, \dot{q}_k</math>, one can use <math>q_{k}, p_{k}</math> as dynamical variables | ||
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==Hamiltonian mechanics== | ==Hamiltonian mechanics== | ||
| − | * conjugate variables are on the equal footing | + | * conjugate variables are on the equal footing |
* [http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조 | * [http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조 | ||
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==Poisson bracket== | ==Poisson bracket== | ||
| − | + | For <math>f(p_i,q_i,t), g(p_i,q_i,t)</math> , we define the Poisson bracket | |
<math>\{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]</math> | <math>\{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]</math> | ||
| 69번째 줄: | 69번째 줄: | ||
<math>\{f,g\} = \frac{1}{i}[u,v]</math> | <math>\{f,g\} = \frac{1}{i}[u,v]</math> | ||
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==phase space== | ==phase space== | ||
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==links and webpages== | ==links and webpages== | ||
| − | * | + | * http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf |
| − | * [http://www.math.ucr.edu/home/baez/classical/ Classical Mechanics] | + | * [http://www.math.ucr.edu/home/baez/classical/ Classical Mechanics] |
** John Baez | ** John Baez | ||
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==question and answers(Math Overflow)== | ==question and answers(Math Overflow)== | ||
| + | * http://mathoverflow.net/questions/30886/applications-of-classical-field-theory | ||
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==history== | ==history== | ||
| 125번째 줄: | 100번째 줄: | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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==related items== | ==related items== | ||
| 136번째 줄: | 111번째 줄: | ||
* [[5 integrable systems and solvable models|integrable Hamiltonian systems and solvable models]] | * [[5 integrable systems and solvable models|integrable Hamiltonian systems and solvable models]] | ||
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==encyclopedia== | ==encyclopedia== | ||
| 154번째 줄: | 129번째 줄: | ||
* [http://en.wikipedia.org/wiki/Action_%28physics%29 http://en.wikipedia.org/wiki/Action_(physics)] | * [http://en.wikipedia.org/wiki/Action_%28physics%29 http://en.wikipedia.org/wiki/Action_(physics)] | ||
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==books== | ==books== | ||
| + | * Classical mechanics [[2610572/attachments/1142452|Classical_Mechanics.djvu]]V.I. Arnold | ||
| + | * [[Emmy Noether’s Wonderful Theorem]] | ||
| + | * [http://library.nu/docs/1U9OCRM7QY/Electrodynamics%20and%20Classical%20Theory%20of%20Fields%20and%20Particles Electrodynamics and Classical Theory of Fields and Particles] | ||
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==expositions== | ==expositions== | ||
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* 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 | ||
[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:physics]] | [[분류:physics]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:classical mechanics]] | [[분류:classical mechanics]] | ||
2013년 4월 21일 (일) 05:44 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
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)
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