Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

scholarly journals Fractional formulation of Podolsky Lagrangian density

2022 ◽  
Vol 9 (2) ◽  
pp. 136-141
Author(s):  
Amer D. Al-Oqali ◽  
Keyword(s):  
Field Theory ◽  
Stress Tensor ◽  
Fractional Derivatives ◽  
Lagrangian Density ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Classical Field Theory ◽  
Higher Order ◽  
Classical Field ◽  
Energy Stress

Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

First steps in classical field theory

2021 ◽  
pp. 435-448
Author(s):  
Andrew M. Steane
Keyword(s):  
Field Theory ◽  
Yukawa Potential ◽  
Classical Field Theory ◽  
Classical Field ◽  
Dirac Spinor ◽  
Dirac Equations ◽  
Flat Spacetime ◽  
Spinor Fields ◽  
The Relationship ◽  
Conserved Currents

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.

scholarly journals A gauge-theoretical approach to elasticity with microrotations

2012 ◽  
Vol 468 (2141) ◽  
pp. 1391-1407 ◽  
Author(s):  
C. G. Böhmer ◽  
Yu. N. Obukhov
Keyword(s):  
Field Theory ◽  
Elasticity Theory ◽  
Theoretical Approach ◽  
Gauge Theories ◽  
Equations Of Motion ◽  
Classical Field Theory ◽  
Classical Field

We formulate elasticity theory with microrotations using the framework of gauge theories, which has been developed and successfully applied in various areas of gravitation and cosmology. Following this approach, we demonstrate the existence of particle-like solutions. Mathematically, this is due to the fact that our equations of motion are of sine-Gordon type and thus have soliton-type solutions. Similar to Skyrmions and Kinks in classical field theory, we can show explicitly that these solutions have a topological origin.

scholarly journals Newton &-vs Lorentz

2018 ◽  
Vol 14 (3) ◽  
pp. 5869-5872
Author(s):  
Armando Tomas Canero
Keyword(s):  
Magnetic Field ◽  
Field Theory ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
The Body ◽  
Circular Path ◽  
Third Law ◽  
A Minor ◽  
The Magnetic Field

Is there a point of divergence between Classical Mechanics and Electromagnetism? This discrepancy is raised by many authors and arises between Newton's third law and the equation of Lorentz forces. Due to the transcendence of these expressions, their wide application in different situations is not a minor issue and should be given a consistent interpretation with both theories. The discrepancy mentioned is based in that: according to the calculations of classical field theory, a particle with an electric charge moving immersed in a magnetic field suffers an action that diverts its trajectory, making it describe a circular path, which can not be compensated through a contrary force in the body that generated the magnetic field. The force on this second body is predicted, by this theory, at ninety degrees from the first, thus contradicting the principle of action and reaction. This study shows why the Lorentz law does not contradict Newton's third law and gives a consistent explanation of how the equations of classical field theory should be applied so that the result is correct.

scholarly journals Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

2016 ◽  
Vol 13 (06) ◽  
pp. 1650072 ◽  
Author(s):  
Václav Zatloukal
Keyword(s):  
Field Theory ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Classical Field Theory ◽  
Classical Field ◽  
Hamiltonian Constraint ◽  
Hamiltonian Mechanics ◽  
Weyl Scalar ◽  
Hamilton Jacobi Equation ◽  
General Method

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton–Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder–Weyl scalar field theory, and string theory.

Stringlike solitons in toroidal coordinates

1979 ◽  
Vol 57 (4) ◽  
pp. 590-592 ◽  
Author(s):  
J. G. Williams
Keyword(s):  
Field Theory ◽  
Equations Of Motion ◽  
Classical Field Theory ◽  
Finite Energy ◽  
Classical Field ◽  
Toroidal Coordinates ◽  
Three Space

A classical field theory is studied in three space dimensions for the case in which the field variables range over a 2-sphere. Toroidal coordinates are found to be the most natural and lead to separation in the equations of motion. A number of finite energy solutions are indicated and correspond to structures extended in space.

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