"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 |
** energy | ** energy | ||
** momentum | ** momentum | ||
| 16번째 줄: | 16번째 줄: | ||
==notation== | ==notation== | ||
| − | * | + | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math> |
* <math>T</math> kinetic energy | * <math>T</math> kinetic energy | ||
* <math>V</math> potential energy | * <math>V</math> potential energy | ||
| − | * | + | * We have Lagrangian <math>L=T-V</math> |
| − | * | + | * 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> is twice of kinetic energy | * <math>p\dot q</math> is twice of kinetic energy | ||
| − | * | + | * Thus the Hamiltonian represents <math>H=T+V</math> the total energy of the system |
| 39번째 줄: | 39번째 줄: | ||
==canonically conjugate momentum== | ==canonically conjugate momentum== | ||
| − | * | + | * 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 | ||
| 50번째 줄: | 50번째 줄: | ||
==Hamiltonian mechanics== | ==Hamiltonian mechanics== | ||
| − | * | + | * 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 | ||
| 132번째 줄: | 132번째 줄: | ||
==expositions== | ==expositions== | ||
| + | * Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences | ||
| + | * Caudrey, P. J., J. C. Eilbeck, and J. D. Gibbon. 1975. “The Sine-Gordon Equation as a Model Classical Field Theory.” Il Nuovo Cimento B Series 11 25 (2) (February 1): 497–512. doi:10.1007/BF02724733. | ||
* Müller, Dr Volkhard F. 1969. “Introduction to the Lagrangian Method.” In Current Algebra and Phenomenological Lagrange Functions, 42–52. Springer Tracts in Modern Physics 118 50. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0045916. | * Müller, Dr Volkhard F. 1969. “Introduction to the Lagrangian Method.” In Current Algebra and Phenomenological Lagrange Functions, 42–52. Springer Tracts in Modern Physics 118 50. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0045916. | ||
| − | + | ||
| − | |||
==books== | ==books== | ||
| − | * | + | * Classical mechanics, V.I. Arnold |
* [[Emmy Noether’s Wonderful Theorem]] | * [[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] |
2013년 12월 16일 (월) 16:48 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
computational resource
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)
expositions
- Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences
- Caudrey, P. J., J. C. Eilbeck, and J. D. Gibbon. 1975. “The Sine-Gordon Equation as a Model Classical Field Theory.” Il Nuovo Cimento B Series 11 25 (2) (February 1): 497–512. doi:10.1007/BF02724733.
- Müller, Dr Volkhard F. 1969. “Introduction to the Lagrangian Method.” In Current Algebra and Phenomenological Lagrange Functions, 42–52. Springer Tracts in Modern Physics 118 50. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0045916.
books
- Classical mechanics, V.I. Arnold
- Emmy Noether’s Wonderful Theorem
- Electrodynamics and Classical Theory of Fields and Particles