Relativistische Verallgemeinerung des Lenzschen Vektors (I) / Relativistic generalisation of the Lenz vector (I)

1984 ◽  
Vol 39 (8) ◽  
pp. 720-732
Author(s):  
Eberhard Kern
Keyword(s):  
Angular Momentum ◽  
Kepler Problem ◽  
Coulomb Field ◽  
Equation Of Motion ◽  
Central Field ◽  
Relativistic Energy ◽  
Geometric Meaning ◽  
Type Curves ◽  
Relativistic Problem ◽  
Charged Nucleus

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

Motion of a particle in three-dimensional fields

2020 ◽  
pp. 6-11
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Angular Momentum ◽  
Zeeman Effect ◽  
Three Dimensional ◽  
Harmonic Oscillation ◽  
Coulomb Field ◽  
Central Field ◽  
Small Perturbations ◽  
Classical Analog ◽  
Small Force ◽  
The Given

In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.

Motion of a particle in three-dimensional fields

2020 ◽  
pp. 91-138
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Angular Momentum ◽  
Zeeman Effect ◽  
Three Dimensional ◽  
Harmonic Oscillation ◽  
Coulomb Field ◽  
Central Field ◽  
Small Perturbations ◽  
Classical Analog ◽  
Small Force ◽  
The Given

In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.

scholarly journals Hammond versus Ford radiation reaction force with the attractive Coulomb field

2018 ◽  
Vol 64 (2) ◽  
pp. 187
Author(s):  
G. Ares de Parga ◽  
S. Domínguez-Hernández ◽  
E. Salinas-Hernández
Keyword(s):  
Reaction Force ◽  
Coulomb Field ◽  
Radiation Reaction ◽  
Equation Of Motion ◽  
Central Field ◽  
Force Term ◽  
Point Particles ◽  
Charged Point ◽  
Relativistic Equations ◽  
Radiation Reaction Force

The classical central field is analyzed within the Hammond theory of radiation reaction force. For the attractive Coulomb field, the trajectories deduced from Ford and Hammond equations are numerically obtained. Ford and Hammond equations are rewritten by using a recent correction to the non-relativistic equations for charged point particles which include a radiation reaction force term. Also, for the attractive Coulomb case, the trajectories are numerically obtained for both corrected equations. A comparison between all these trajectories is made. It is proved that Hammond equation satisfies the constraint proposed by Dirac of getting an equation of motion which should make the electron in the hydrogen atom spiralling inwards and ultimately falling into the nucleus. A further analysis of the applicability of such a theory is described for experiments particularly in Plasma Physics and some comments are made for the generalization of Hammond equation to General Relativity.

scholarly journals The Relativistic and the Hidden Momentum of Minkowski and Abraham in Relativistic Energy Wave

2021 ◽  
Author(s):  
Daniel Cardoso
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Mechanical Energy ◽  
Recent Theory ◽  
Translational Energy ◽  
Angle Of Incidence ◽  
Relativistic Energy ◽  
Conservation Of Energy ◽  
Ignition Device

An analysis of the consistency of the Abraham and Minkowski momenta in the determination of the photon trajectory was carried out considering a new principle of conservation of the photon's mechanical energy, in which the photon conserves translational energy in orbital angular momentum when transiting between two media, introducing the relativistic energy wave (REW). The confrontation between REW and the recent theory of space-time waves (ST) was considered, pondering your differences. Throughout this study it was possible to verify that the Abraham momentum appears a relativistic photon ignition device in the transition between two media, acting as the hidden momentum of the Minkowski’s relativistic momentum. The wavy behavior in the matter is relativistic, and the relativistic trajectory appears with delays and advances, with points of synchronization between source-observer. The classical or relativistic trajectories are determined as a function of the angle of incidence and the relative refractive index, by one of two distinct non-additive torques, the classic by Abraham or the relativistic by Minkowski. It was found that the same analysis conducted under the principle of conservation of the mechanical energy of the photon can be treated by an new Doppler, Relativistic Apparent, that can be confused with other Dopplers in the treatment of redshift from distant sources. It was found that the conservation of energy in Orbital Angular Momentum (OAM), in the interaction with matter, explains that the synchronization instants are found in the inversion of the OAM, where the advances and delays of REW occur under negligible variations of the OAM, however, opposites.

Ein klassisches Teilchenmodell mit variablem Gravitationskoeffizienter

1962 ◽  
Vol 17 (7) ◽  
pp. 554-558
Author(s):  
Jochen Lindner
Keyword(s):  
Coulomb Field ◽  
Coulomb Force ◽  
Equation Of Motion ◽  
Great Distance ◽  
Classical Particle ◽  
Unified Theory ◽  
Specific Charge ◽  
Field Function ◽  
Typical Function ◽  
Theory Of Gravitation

The unified theory of gravitation and the electromagnetic field in the form suggested by BECHERT 1 has solutions which correspond to the model of a classical particle of mass Moo and charge Q. We shall assume that the coefficient of gravitation χ is not a constant but a field function. The equation of motion is derived for this case. It shows that a suitable choice of the field function χ leads to a correct COULOMB field as well as to a correct gravitational field (corresponding to Q and Mo) in great distance from the particle. The extension of the particle is characterized by the classical radius L=Q2/Moc2 of the particle, it holds together by the balance between COULOMB force and gravitation. The specific charge turns out to be a typical function of the distance from the center of the particle.

INFELD FACTORIZATION AND ANGULAR MOMENTUM

1956 ◽  
Vol 34 (4) ◽  
pp. 343-349 ◽  
Author(s):  
H. R. Coish
Keyword(s):  
Angular Momentum ◽  
Spherical Harmonics ◽  
Momentum Space ◽  
Kepler Problem ◽  
Free Particle ◽  
Magnetic Pole ◽  
Pole System

The connection between Infeld factorization operators and angular momentum operators, well known for spherical harmonics, is extended to other factorization problems by explicitly recognizing them as angular momentum problems. These other problems are: the symmetric top, electron-magnetic pole system, Weyl's spherical harmonics with spin, free particle on a hypersphere. The Kepler problem is also included for it may be thrown into the form of a four-dimensional angular momentum problem. The transformation to momentum space for this problem is very much simplified by the connection between Infeld factorization and angular momentum.

scholarly journals Bakerian Lecture: Nuclear constitution of atoms

1920 ◽  
Vol 97 (686) ◽  
pp. 374-400 ◽  
Keyword(s):  
Large Mass ◽  
Atomic Weight ◽  
Single Atom ◽  
Central Field ◽  
X Rays ◽  
Hyperbolic Orbit ◽  
Series Of Experiments ◽  
Charged Nucleus ◽  
Positively Charged ◽  
Α Particles

Introduction .—The conception of the nuclear constitution of atoms arose initially from attempts to account for the scattering of α-particles through large angles in traversing thin sheets of matter. Taking into account the large mass and velocity of the α-particles, these large deflexions were very remarkable, and indicated that very intense electric or magnetic fields exist within the atom. To account for these results, it was found necessary to assume that the atom consists of a charged massive nucleus of dimensions very small compared with the ordinarily accepted magnitude of the diameter of the atom. This positively charged nucleus contains most of the mass of the atom, and is surrounded at a distance by a distribution of negative electrons equal in number to the resultant positive charge on the nucleus. Under these conditions, a very intense electric field exists close to the nucleus, and the large deflexion of the α-particle in an encounter with a single atom happens when the particle passes close to the nucleus. Assuming that the electric forces between the α-particle and the nucleus varied according to an inverse square law in the region close to the nucleus, the writer worked out the relations connecting the number of α-particles scattered through any angle with the charge on the nucleus and the energy of the α-particle. Under the central field of force, the α-particle describes a hyperbolic orbit round the nucleus, and the magnitude of the deflection depends on the closeness of approach to the nucleus. From the data of scattering of α-particles then available, it was deduced that the resultant charge on the nucleus was about ½ A e , where A is the atomic weight and e the fundamental unit of charge. Geiger and Marsden made an elaborate series of experiments to test the correctness of the theory, and confirmed the main conclusions. They found the nucleus charge was about ½ A e , but, from the nature of the experiments, it was difficult to fix the actual value within about 20 per cent. C. G. Darwin worked out completely the deflexion of the α-particle and of the nucleus, taking into account the mass of the latter, and showed that the scattering experiments of Geiger and Marsden could not be reconciled with any law of central force, except the inverse square. The nuclear constitution of the atom was thus very strongly supported by the experiments on scattering of α-rays. Since the atom is electrically neutral, the number of external electrons surrounding the nucleus must be equal to the number of units of resultant charge on the nucleus. It should be noted that, from the consideration of the scattering of X-rays by light elements, Barkla had shown, in 1911, that the number of electrons was equal to about half the atomic weight. This was deduced from the theory of scattering of Sir J. J. Thomson, in which it was assumed that each of the external electrons in an atom acted as an independent scattering unit.

scholarly journals Elucidating the Nature of the Fine Structure Constant and Indicating the Existence of an Unknown Angular Momentum

2017 ◽  
Vol 9 (4) ◽  
pp. 17
Author(s):  
Koshun Suto
Keyword(s):  
Angular Momentum ◽  
Fine Structure ◽  
Hydrogen Atom ◽  
Structure Constant ◽  
Physical Constant ◽  
Natural World ◽  
Fine Structure Constant ◽  
Relativistic Energy ◽  
Planck Constant ◽  
Virtual Particle

In this paper, the author searches for a formula different from the existing formula in order to elucidate the nature of the fine structure constant a. The relativistic energy of the electron in a hydrogen atom is expressed as E_re,n and the momentum corresponding to that energy is taken to be P_re,n. Also, P_p,n is assumed to be the momentum of a photon emitted when an electron that has been stationary in free space transitions to the inside of a hydrogen atom. When n=1, the ratio of P_re,1 and P_p,1 matches with a. That is, P_p,1/Pre,1=a Also, the formula for the energy of a photon is E=hv. However, this formula has no constant of proportionality. If one wishes to claim that the energy of a photon varies in proportion to the photon's frequency, then a formula containing a constant of proportionality is necessary. Thus, this paper predicts that, in the natural world, there is a minimum unit of angular momentum h_vp smaller than the Planck constant. (The vp in h_vp stands for “virtual particle.”)If this physical constant is introduced, then the formula for the energy of the photon can be written as E=h_vp v/a. If h_vp exists, a formula can also be obtained which helps to elucidate the nature of the fine structure constant.

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