Classical field theory and classical mechanics

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

• 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

\(\{f,g\} = \frac{1}{i}[u,v]\)

links and webpages

• http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf
• 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=

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/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

• 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

• 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.

books

• Classical mechanics, V.I. Arnold
• Emmy Noether’s Wonderful Theorem
• Electrodynamics and Classical Theory of Fields and Particles