"Classical field theory and classical mechanics"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) (→메타데이터) |
|||
(사용자 3명의 중간 판 43개는 보이지 않습니다) | |||
1번째 줄: | 1번째 줄: | ||
− | + | ==introduction== | |
− | * can be formulated using classical fields and | + | * can be formulated using classical fields and Lagrangian density |
* change the coordinates and fields accordingly | * change the coordinates and fields accordingly | ||
* 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 | ||
** angular momentum | ** angular momentum | ||
− | + | ||
− | + | ||
− | + | ==notation== | |
− | * <math> | + | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math> |
− | * <math> | + | * <math>T</math> kinetic energy |
− | * | + | * <math>V</math> potential energy |
− | * | + | * We have Lagrangian <math>L=T-V</math> |
− | * <math>H = | + | * Define the Hamiltonian |
− | * <math>p\dot q</math> | + | * <math>H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L</math> |
− | * | + | * <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 | ||
− | + | ||
− | + | ||
− | + | ==Lagrangian formalism== | |
− | + | * [[Lagrangian formalism]] | |
− | + | ||
− | |||
− | |||
− | + | ||
− | + | ==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 | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ==Hamiltonian mechanics== | |
− | * | + | * conjugate variables are on the equal footing |
− | + | * [http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조 | |
− | |||
− | + | ||
− | + | ||
− | + | ||
− | + | ==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> | ||
93번째 줄: | 69번째 줄: | ||
<math>\{f,g\} = \frac{1}{i}[u,v]</math> | <math>\{f,g\} = \frac{1}{i}[u,v]</math> | ||
− | + | ||
− | |||
− | |||
− | |||
− | |||
− | + | ||
− | + | ==phase space== | |
− | + | ||
− | + | ||
− | + | ==links and webpages== | |
− | + | * http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf | |
+ | * [http://www.math.ucr.edu/home/baez/classical/ Classical Mechanics] | ||
+ | ** John Baez | ||
− | + | ||
− | + | ==question and answers(Math Overflow)== | |
+ | * http://mathoverflow.net/questions/30886/applications-of-classical-field-theory | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ==history== | |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
− | + | ||
− | |||
− | |||
− | + | ||
− | + | ==related items== | |
− | |||
− | |||
− | + | * [[Electromagnetics]] | |
+ | * [[Einstein field equation]] | ||
+ | * [[symplectic geometry]] | ||
+ | * [[Integrable systems and solvable models]] | ||
− | + | ||
+ | ==computational resource== | ||
+ | * https://docs.google.com/file/d/0B8XXo8Tve1cxeWN6Q2pyaE1ZMjg/edit | ||
+ | |||
− | + | ==encyclopedia== | |
− | |||
* http://en.wikipedia.org/wiki/Classical_field_theory | * http://en.wikipedia.org/wiki/Classical_field_theory | ||
* http://en.wikipedia.org/wiki/Continuity_equation | * http://en.wikipedia.org/wiki/Continuity_equation | ||
* http://en.wikipedia.org/wiki/current_density | * http://en.wikipedia.org/wiki/current_density | ||
* [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem] | * [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem] | ||
− | |||
* [http://ko.wikipedia.org/wiki/%EC%9E%91%EC%9A%A9 http://ko.wikipedia.org /wiki/작용] | * [http://ko.wikipedia.org/wiki/%EC%9E%91%EC%9A%A9 http://ko.wikipedia.org /wiki/작용] | ||
* http://en.wikipedia.org/wiki/Canonical_coordinates | * http://en.wikipedia.org/wiki/Canonical_coordinates | ||
152번째 줄: | 127번째 줄: | ||
* http://en.wikipedia.org/wiki/Lagrangian | * http://en.wikipedia.org/wiki/Lagrangian | ||
* http://en.wikipedia.org/wiki/poisson_bracket | * http://en.wikipedia.org/wiki/poisson_bracket | ||
− | * | + | * http://en.wikipedia.org/wiki/Action_(physics) |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | ==books== | |
+ | * Classical mechanics, 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] | ||
− | |||
− | + | ==expositions== | |
+ | * McLachlan, Robert I., Klas Modin, and Olivier Verdier. “Symmetry Reduction for Central Force Problems.” arXiv:1512.04631 [math-Ph], December 14, 2015. http://arxiv.org/abs/1512.04631. | ||
+ | * Nolte, David D. ‘The Tangled Tale of Phase Space’. Physics Today, 2010. http://works.bepress.com/ddnolte/2. | ||
+ | * De León, M., M. Salgado, and S. Vilariño. “Methods of Differential Geometry in Classical Field Theories: K-Symplectic and K-Cosymplectic Approaches.” arXiv:1409.5604 [math-Ph], September 19, 2014. http://arxiv.org/abs/1409.5604. | ||
+ | * 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. | ||
− | * http://arxiv.org/ | + | ==articles== |
+ | * Sebastián Ferraro, Manuel de León, Juan Carlos Marrero, David Martín de Diego, Miguel Vaquero, On the Geometry of the Hamilton-Jacobi Equation and Generating Functions, arXiv:1606.00847 [math-ph], June 02 2016, http://arxiv.org/abs/1606.00847 | ||
+ | * Solanpää, Janne, Perttu Luukko, and Esa Räsänen. ‘Bill2d - a Software Package for Classical Two-Dimensional Hamiltonian Systems’. arXiv:1506.06917 [physics], 23 June 2015. http://arxiv.org/abs/1506.06917. | ||
+ | * Zelikin, Mikhail. “The Fractal Theory of the Saturn Ring.” arXiv:1506.02908 [math-Ph], June 9, 2015. http://arxiv.org/abs/1506.02908. | ||
+ | * Gay-Balmaz, François, and Tudor S. Ratiu. 2014. “A New Lagrangian Dynamic Reduction in Field Theory.” arXiv:1407.0263 [math-Ph], July. http://arxiv.org/abs/1407.0263. | ||
+ | * Sławianowski, J. J., Jr Schroeck, and A. Martens. “Why Must We Work in the Phase Space?” arXiv:1404.2588 [math-Ph], April 4, 2014. http://arxiv.org/abs/1404.2588. | ||
− | |||
− | |||
− | + | [[분류:개인노트]] | |
+ | [[분류:physics]] | ||
+ | [[분류:math and physics]] | ||
+ | [[분류:classical mechanics]] | ||
+ | [[분류:migrate]] | ||
− | * [ | + | ==메타데이터== |
− | * [ | + | ===위키데이터=== |
− | + | * ID : [https://www.wikidata.org/wiki/Q2603912 Q2603912] | |
− | + | ===Spacy 패턴 목록=== | |
+ | * [{'LOWER': 'classical'}, {'LOWER': 'field'}, {'LEMMA': 'theory'}] |
2021년 2월 17일 (수) 02:21 기준 최신판
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)
books
- Classical mechanics, V.I. Arnold
- Emmy Noether’s Wonderful Theorem
- Electrodynamics and Classical Theory of Fields and Particles
expositions
- McLachlan, Robert I., Klas Modin, and Olivier Verdier. “Symmetry Reduction for Central Force Problems.” arXiv:1512.04631 [math-Ph], December 14, 2015. http://arxiv.org/abs/1512.04631.
- Nolte, David D. ‘The Tangled Tale of Phase Space’. Physics Today, 2010. http://works.bepress.com/ddnolte/2.
- De León, M., M. Salgado, and S. Vilariño. “Methods of Differential Geometry in Classical Field Theories: K-Symplectic and K-Cosymplectic Approaches.” arXiv:1409.5604 [math-Ph], September 19, 2014. http://arxiv.org/abs/1409.5604.
- 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.
articles
- Sebastián Ferraro, Manuel de León, Juan Carlos Marrero, David Martín de Diego, Miguel Vaquero, On the Geometry of the Hamilton-Jacobi Equation and Generating Functions, arXiv:1606.00847 [math-ph], June 02 2016, http://arxiv.org/abs/1606.00847
- Solanpää, Janne, Perttu Luukko, and Esa Räsänen. ‘Bill2d - a Software Package for Classical Two-Dimensional Hamiltonian Systems’. arXiv:1506.06917 [physics], 23 June 2015. http://arxiv.org/abs/1506.06917.
- Zelikin, Mikhail. “The Fractal Theory of the Saturn Ring.” arXiv:1506.02908 [math-Ph], June 9, 2015. http://arxiv.org/abs/1506.02908.
- Gay-Balmaz, François, and Tudor S. Ratiu. 2014. “A New Lagrangian Dynamic Reduction in Field Theory.” arXiv:1407.0263 [math-Ph], July. http://arxiv.org/abs/1407.0263.
- Sławianowski, J. J., Jr Schroeck, and A. Martens. “Why Must We Work in the Phase Space?” arXiv:1404.2588 [math-Ph], April 4, 2014. http://arxiv.org/abs/1404.2588.
메타데이터
위키데이터
- ID : Q2603912
Spacy 패턴 목록
- [{'LOWER': 'classical'}, {'LOWER': 'field'}, {'LEMMA': 'theory'}]