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6 results

  • 12 - An Excursion: Constrained Dynamics

  • from Part III - Canonical Field Theories with Gravity
  • Piljin Yi, Korea Institute for Advanced Study
  • Book:
    Gravitation for Theorists
    Published online:
    01 March 2026
    Print publication:
    19 March 2026, pp 235-254

  • Summary

    In preparation for the ADM formulation of General Relativity, we quickly scan Dirac's theory of constrained systems. How to deal with dynamics when the number of variables is larger than the true degrees of freedom is at issue. Starting from a familiar classical mechanics with Lagrange multipliers, we classify constraints into the first class and the second class. The former is particularly relevant for field theories with gauge redundancies, as is the case with General Relativity. Again, the Maxwell theory is invoked as a prototype, with the Gauss constraint given a unique meaning as the generator of the gauge redundancy.

  • 9 - Poisson Brackets for Field Theory and Equations of Motion: Applications

  • from Part I - General Properties of Fields; Scalars and Gauge Fields
  • Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
  • Book:
    Classical Field Theory
    Published online:
    04 March 2019
    Print publication:
    14 March 2019, pp 81-88

  • Summary

    We examine Poisson brackets in field theory and the symplectic formulation of Hamiltonian dynamics. We start by describing the symplectic formulation of classical mechanics. Then we generalize it and Poisson brackets to field theory. As examples of the formalism, we consider a scalar field with canonical kinetic term and the nonlinear sigma model.

  • 1 - Short Review of Classical Mechanics

  • from Part I - General Properties of Fields; Scalars and Gauge Fields
  • Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
  • Book:
    Classical Field Theory
    Published online:
    04 March 2019
    Print publication:
    14 March 2019, pp 3-13

  • Summary

    In the first chapter, the most important concepts of classical mechanics are quickly reviewed. The Lagrangian and Hamiltonian formalism are described. The way to deal with systems with constraints is described. Poisson brackets and the use of canonical transformations in the Hamiltonian formalism, as well as the basics of Hamilton–Jacobi theory complete this chapter.

  • Poisson Brackets with Prescribed Casimirs

  • Pantelis A. Damianou, Fani Petalidou
  • Journal:
    Canadian Journal of Mathematics / Volume 64 / Issue 5 / 01 October 2012
    Published online by Cambridge University Press:
    20 November 2018, pp. 991-1018
    Print publication:
    01 October 2012
    • Article
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  • We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$ , while, in the case where $M$ is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.