Classical field theory and classical mechanics

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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.

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