"Classical field theory and classical mechanics"의 두 판 사이의 차이
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| 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_(physics) |
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| + | ==expositions== | ||
| + | * 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. | ||
| + | * Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences | ||
| 136번째 줄: | 141번째 줄: | ||
* [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] | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:physics]] | [[분류:physics]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:classical mechanics]] | [[분류:classical mechanics]] | ||
2013년 12월 11일 (수) 11:07 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
- 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.
- Benci V. Fortunato D., Solitary waves in classical field theory, in Nonlinear Analysis and Applications to Physical Sciences
books
- Classical mechanics Classical_Mechanics.djvuV.I. Arnold
- Emmy Noether’s Wonderful Theorem
- Electrodynamics and Classical Theory of Fields and Particles