"Classical field theory and classical mechanics"의 두 판 사이의 차이

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imported>Pythagoras0
 
(사용자 2명의 중간 판 14개는 보이지 않습니다)
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
16번째 줄: 16번째 줄:
 
==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> 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>
+
* 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> 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
+
* 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>
+
* 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
+
* 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
  
106번째 줄: 106번째 줄:
 
==related items==
 
==related items==
  
−
* [[Electromagnetics|Electromagnetism]]
+
* [[Electromagnetics]]
−
* [[Einstein field equation|Einstein field hequation]]
+
* [[Einstein field equation]]
−
* [[symplectic geometry|sympletic geometry]]
+
* [[symplectic geometry]]
−
* [[5 integrable systems and solvable models|integrable Hamiltonian systems and solvable models]]
+
* [[Integrable systems and solvable models]]
  
 
   
 
   
−
 
+
==computational resource==
 +
* https://docs.google.com/file/d/0B8XXo8Tve1cxeWN6Q2pyaE1ZMjg/edit
 
   
 
   
  
121번째 줄: 122번째 줄:
 
* 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
127번째 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_%28physics%29 http://en.wikipedia.org/wiki/Action_(physics)]
+
* http://en.wikipedia.org/wiki/Action_(physics)
  
−
 
  
 
==books==
 
==books==
−
* Classical mechanics [[2610572/attachments/1142452|Classical_Mechanics.djvu]]V.I. Arnold
+
* 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]
+
* [http://library.nu/docs/1U9OCRM7QY/Electrodynamics%20and%20Classical%20Theory%20of%20Fields%20and%20Particles Electrodynamics and Classical Theory of Fields and Particles]
  
−
 
−
 
−
 
  
 
==expositions==
 
==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
 
* 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.
 +
 +
 +
 
[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[분류:physics]]
 
[[분류:physics]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:classical mechanics]]
 
[[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




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



related items


computational resource


encyclopedia


books


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

Spacy 패턴 목록

  • [{'LOWER': 'classical'}, {'LOWER': 'field'}, {'LEMMA': 'theory'}]
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