"Classical field theory and classical mechanics"의 두 판 사이의 차이
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| 20번째 줄: | 20번째 줄: | ||
* We have Lagrangian <math>L=T-V</math><br> | * We have Lagrangian <math>L=T-V</math><br> | ||
* Define the Hamiltonian<br> | * Define the Hamiltonian<br> | ||
| − | * <math>H =p\dot q-L</math><br> | + | * <math>H =\sum_{k=1}^{N} p\dot q-L</math><br> |
* <math>p\dot q</math> is twice of kinetic energy<br> | * <math>p\dot q</math> is twice of kinetic energy<br> | ||
* Thus the Hamiltonian represents <math>H=T+V</math> the total energy of the system<br> | * Thus the Hamiltonian represents <math>H=T+V</math> the total energy of the system<br> | ||
| 35번째 줄: | 35번째 줄: | ||
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| + | <h5 style="margin: 0px; line-height: 2em;">canonically conjugate momentum</h5> | ||
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| + | * dynamical variables <math>q_{k}, \dot{q}_k</math> for <math>k=1,\cdots, n</math><br> | ||
| + | * canonically conjugate momenta<br><math>p_{k}=\frac{\partial L}{\partial \dot{q}_k}</math><br> | ||
| + | * instead of <math>q_{k}, \dot{q}_k</math>, one can use <math>q_{k}, p_{k}</math> | ||
| 64번째 줄: | 70번째 줄: | ||
<h5 style="margin: 0px; line-height: 2em;">phase space</h5> | <h5 style="margin: 0px; line-height: 2em;">phase space</h5> | ||
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2012년 6월 10일 (일) 17:23 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
- \(T\) kinetic energy
- \(V\) potential energy
- We have Lagrangian \(L=T-V\)
- Define the Hamiltonian
- \(H =\sum_{k=1}^{N} p\dot q-L\)
- \(p\dot q\) is twice of kinetic energy
- Thus the Hamiltonian represents \(H=T+V\) the total energy of the system
Lagrangian formalism
canonically conjugate momentum
- dynamical variables \(q_{k}, \dot{q}_k\) for \(k=1,\cdots, n\)
- canonically conjugate momenta
\(p_{k}=\frac{\partial L}{\partial \dot{q}_k}\) - instead of \(q_{k}, \dot{q}_k\), one can use \(q_{k}, p_{k}\)
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
하위페이지
links and webpages
question and answers(Math Overflow)
history
- Electromagnetism
- Einstein field hequation
- sympletic geometry
- integrable Hamiltonian systems and solvable models
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)
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- Classical mechanics Classical_Mechanics.djvuV.I. Arnold
- Electrodynamics and Classical Theory of Fields and Particles