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

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<h5>introduction</h5>
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==introduction==
  
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* can be formulated using classical fields and lagrangian density
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* 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
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* three important conserved quantity
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** energy
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** momentum
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** angular momentum
  
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<h5>Euler-Lagrange equation</h5>
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==notation==
  
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* if field satisfies the equation of motion, EL is satisfied<br><math>\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0.</math><br>
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* dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, N</math>
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* <math>T</math> kinetic energy
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* <math>V</math> potential energy
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* We have Lagrangian <math>L=T-V</math>
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* Define the Hamiltonian
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* <math>H =\sum_{k=1}^{N} p_{k}\dot{q}_{k}-L</math>
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* <math>p\dot q</math> is twice of kinetic energy
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* Thus the Hamiltonian represents <math>H=T+V</math> the total energy of the system
  
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<h5>equation of continuity</h5>
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==Lagrangian formalism==
  
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* current density <math>J_{\mu}=(J_0,J_1,J_2,J_3)</math> satisfies<br><math>\partial^{\mu} J_{\mu}=0</math><br>
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* [[Lagrangian formalism]]
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*  we get a conserved quantity<br><math>G=\int_V J_0(x) \,d^3 x</math><br>
 
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* Lagrangian can be used to express the current density explicity
 
  
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==canonically conjugate momentum==
  
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<h5>currents</h5>
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* canonically conjugate momenta<math>p_{k}=\frac{\partial L}{\partial \dot{q}_k}</math>
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* 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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* quantum analogues of the conser
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<h5>history</h5>
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==Hamiltonian mechanics==
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* conjugate variables are on the equal footing
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* [http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조
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==Poisson bracket==
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For <math>f(p_i,q_i,t), g(p_i,q_i,t)</math> , we define the Poisson bracket
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<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>
  
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In quantization we have correspondence
  
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<math>\{f,g\} = \frac{1}{i}[u,v]</math>
  
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<h5>related items</h5>
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* [[Electromagnetics|Electromagnetism]]
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==phase space==
  
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<h5>books</h5>
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==links and webpages==
  
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* [[4909919|찾아볼 수학책]]<br>
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* http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf
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* http://gigapedia.info/1/
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* [http://www.math.ucr.edu/home/baez/classical/ Classical Mechanics]
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* http://gigapedia.info/1/
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** John Baez
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
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<h5>encyclopedia</h5>
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==question and answers(Math Overflow)==
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* http://mathoverflow.net/questions/30886/applications-of-classical-field-theory
  
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* http://ko.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/Classical_field_theory
 
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* http://en.wikipedia.org/wiki/Continuity_equation
 
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* http://en.wikipedia.org/wiki/current_density
 
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* [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem]
 
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* http://en.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
  
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<h5>question and answers(Math Overflow)</h5>
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==history==
  
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* http://mathoverflow.net/search?q=
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
  
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<h5>blogs</h5>
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==related items==
  
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* 구글 블로그 검색<br>
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* [[Electromagnetics]]
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** http://blogsearch.google.com/blogsearch?q=
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* [[Einstein field equation]]
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** http://blogsearch.google.com/blogsearch?q=
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* [[symplectic geometry]]
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** http://blogsearch.google.com/blogsearch?q=
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* [[Integrable systems and solvable models]]
  
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxeWN6Q2pyaE1ZMjg/edit
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==encyclopedia==
  
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<h5>articles</h5>
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* http://en.wikipedia.org/wiki/Classical_field_theory
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* http://en.wikipedia.org/wiki/Continuity_equation
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* http://en.wikipedia.org/wiki/current_density
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* [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem]
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* [http://ko.wikipedia.org/wiki/%EC%9E%91%EC%9A%A9 http://ko.wikipedia.org /wiki/작용]
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* http://en.wikipedia.org/wiki/Canonical_coordinates
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* http://en.wikipedia.org/wiki/Lagrangian_mechanics
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* http://en.wikipedia.org/wiki/Lagrangian
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* http://en.wikipedia.org/wiki/poisson_bracket
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* http://en.wikipedia.org/wiki/Action_(physics)
  
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* [[2010년 books and articles|논문정리]]
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==books==
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* http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
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* Classical mechanics, V.I. Arnold
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* http://www.ams.org/mathscinet
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* [[Emmy Noether’s Wonderful Theorem]]
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* http://www.zentralblatt-math.org/zmath/en/
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* [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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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://dx.doi.org/
 
  
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==expositions==
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* 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.
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* Nolte, David D. ‘The Tangled Tale of Phase Space’. Physics Today, 2010. http://works.bepress.com/ddnolte/2.
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* 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.
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* Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences
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* 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.
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* 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.
  
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<h5>experts on the field</h5>
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==articles==
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* 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
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* 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.
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* Zelikin, Mikhail. “The Fractal Theory of the Saturn Ring.” arXiv:1506.02908 [math-Ph], June 9, 2015. http://arxiv.org/abs/1506.02908.
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* 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.
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* 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.
  
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* http://arxiv.org/
 
  
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[[분류:개인노트]]
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[[분류:physics]]
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[[분류:math and physics]]
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[[분류:classical mechanics]]
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[[분류:migrate]]
  
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<h5>TeX </h5>
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q2603912 Q2603912]
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===Spacy 패턴 목록===
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* [{'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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