Canonical solutions of variational problems and canonical equations of mechanics

Mekhitarian, V. M.

doi:10.3103/s1068337213010015

Cited by 4 publications

(4 citation statements)

References 5 publications

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“…In [28], to construct the relativistic theory on the basis of variational principles, the canonical (non-parametric) solutions of the variational problem for canonically defined integral functionals have been considered and the canonical solutions of the variational problems of mechanics in the Minkowski spaces are written. Because of unifying the variational principles of least action, flow, and hyperflow, the canonically invariant equations for the generalized momentum are obtained.…”

Section: Figurementioning

confidence: 99%

Equations of Relativistic and Quantum Mechanics (without Spin)

Mekhitarian¹

2020

Quantum Mechanics

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A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.

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Section: Figurementioning

confidence: 99%

Equations of Relativistic and Quantum Mechanics (without Spin)

Mekhitarian¹

2020

Quantum Mechanics

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“…Since the spin of the field has been determined locally, its direction from point to point can change, provided that the internal structure is preserved. The spinor field, as in the general case of any vector potential A ¼ φΩ, must satisfy the basic equations for the fields [2].…”

Section: Representation Of the Spinor Fields In Minkowski Spacesmentioning

confidence: 99%

“…The material presented in this chapter is based on the new approaches of relativistic and quantum mechanics developed in the works [1][2][3][4]. Equations, which are obtained by applying the invariance principle for the total four-dimensional momentum of the system "field + particle," have some significant advantages as compared with its analog equations such as the Klein-Fock-Gordon and Dirac equations.…”

Section: Introductionmentioning

confidence: 99%

Spinor Fields

Mekhitarian¹

2023

Recent Topics and Innovations in Quantum Field Theory

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A spinor representation of the generalized energy-momentum density 4-vector is proposed, and examples of such representations for various particles and fields are given. This representation corresponds to the classical representation of the particle’s own rotation, which is described by the diagonal matrix of the moment of inertia. The concept of self-angular rotation of a particle is defined as a spatial characteristic of the field, at each point of which there is a local vortex rotation with an angular velocity Ω – a spinor field. The matrix representation of the vortex rotation Ω (spinor) and the values of the components of such a representation are derived from the matrix representation of the Lorentz transformation. The traditional concept of spin-orbit interaction, as the interaction of the magnetic moment of a particle with the magnetic field of orbital motion, is presented as the interaction of a charged particle with a spinor field. Solutions to the problems of particle motion in an external spinor field in the case of a hydrogen-like atom and planetary motion, splitting of the electron energy levels of an atom in an external magnetic field, deflection of a photon by the gravitational field, and representations in metric spaces are presented.

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“…No doubt, the results obtained here should be considered and verified within the approaches based on the fundamental principles of modern physics. Such general and fundamental approaches, which take into account the dependence of the interaction on the motion velocity of a particle in an external field are presented in [23,24]. A specific justification of the Faraday law of electromagnetic induction and the derivation of expression for the force acting on a moving charge in an external electromagnetic field within the framework of these approaches will be discussed in a subsequent paper.…”

Section: Induction Current and The Heating Of Mediummentioning

confidence: 99%

The Faraday law of induction for an arbitrarily moving charge

Mekhitarian

2016

J. Contemp. Phys.

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Abstract⎯The Faraday law of electromagnetic induction for an arbitrarily moving charge is generalized and the expression for the force acting on the charge in an alternating magnetic field is obtained. It is shown that besides the Lorentz force perpendicular to the velocity of the particle, the Faraday force parallel to the particle velocity and proportional to it is acting on the charge, too. The equations of motion of the charged particle and the magnetic moment are obtained in the time-varying magnetic field. The problems of induction acceleration of charged particles (betatron) and induction heating of medium (plasma, plasma betatron) are considered.

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Canonical solutions of variational problems and canonical equations of mechanics

Cited by 4 publications

References 5 publications

Equations of Relativistic and Quantum Mechanics (without Spin)

Equations of Relativistic and Quantum Mechanics (without Spin)

Spinor Fields

The Faraday law of induction for an arbitrarily moving charge

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