scholarly journals Relativistic equations for elementary particles

1948 ◽  
Vol 192 (1029) ◽  
pp. 195-218 ◽  
Keyword(s):  
Elementary Particle ◽  
Wave Equation ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Wave Form ◽  
Total Charge ◽  
Field Equations ◽  
Relativistic Approximation ◽  
Special Cases ◽  
Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

On a non-symmetric theory of the pure gravitational field

1958 ◽  
Vol 54 (1) ◽  
pp. 72-80 ◽  
Author(s):  
D. W. Sciama
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Symmetric Tensor ◽  
Momentum Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Unified Field

ABSTRACTIt is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

Modeling, Simulation, and Modal Analysis Hydraulic Valve Lifter With Oil Compressibility Effects

1989 ◽  
Author(s):  
T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Check Valve ◽  
Linear Differential Equations ◽  
Third Order ◽  
Mode Behavior ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

Abstract A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multi-mode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid In the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, non-linear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

Modeling, Simulation, and Modal Analysis of a Hydraulic Valve Lifter With Oil Compressibility Effects

1991 ◽  
Vol 113 (1) ◽  
pp. 46-54 ◽  
Author(s):  
C. T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Check Valve ◽  
Third Order ◽  
Bulk Oil ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multimode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid in the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, nonlinear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

The Two-Dimensional Electrostatic Solutions of Born's new Field Equations

1935 ◽  
Vol 31 (1) ◽  
pp. 50-68 ◽  
Author(s):  
M. H. L. Pryce
Keyword(s):  
General Solution ◽  
General Theory ◽  
Singular Solution ◽  
Equations Of Motion ◽  
Constant Field ◽  
Two Dimensional ◽  
Field Equations ◽  
Special Cases

The general solution of Born's new field equations is found for the two-dimensional electrostatic case, by which the coordinates are expressed as functions of the field vectors, Conditions for inversion are discussed. Special cases are worked out, namely: singnle charge, two charges, charge in a constant field. Expressions are given for forces acting on the charges. A singular solution is also discussed, with reference to the neutron. The implication of the solutions on the general theory and the equations of motion is discussed in the conclusion.

scholarly journals Particle paths of general relativity as geodesics of an affine connection

1970 ◽  
Vol 3 (3) ◽  
pp. 325-335 ◽  
Author(s):  
R. Burman
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Momentum Tensor ◽  
Field Equations ◽  
Test Particles ◽  
Special Cases ◽  
Particle Paths ◽  
Arbitrary Energy

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

scholarly journals EN The wave model of secondary flows and coherent structures in pipes

2020 ◽  
Vol 55 (5-6) ◽  
pp. 273-281
Author(s):  
S. Surkov
Keyword(s):  
Wave Equation ◽  
Stream Function ◽  
Coherent Structures ◽  
Large Scale ◽  
Experimental Studies ◽  
Viscous Friction ◽  
Orthogonal Decomposition ◽  
Secondary Flows ◽  
Friction Forces ◽  
Special Cases

In this article, a theoretical analysis of the flows arising in the cross sections of fluid and gas flows is performed. Such flows are subdivided into secondary flows and coherent structures. From experimental studies it is known that both types of flows are long-lived large-scale movements (LSM) stretched along the flow. The relative stability of the vortices is traditionally explained by the fact that the viscous friction forces that inhibit the rotation are compensated by the intensification of the swirl when moving slowly rotating peripheral layers to the center of the vortex due to longitudinal tension. An analysis of this mechanism made it possible to develop a relatively simple model of vortex structures in which the viscous friction forces and axial expansion are considered to be infinitesimal. Under these assumptions, one can use the equations of motion of an ideal fluid in the variables “stream function - vorticity”. It is shown that under certain assumptions these equations take the form of a wave equation, and the boundary conditions are the condition that the stream function on the solid walls of the flow equals zero. The obtained solutions of the wave equation describe the following special cases: Goertler’s vortices between rotating cylinders, secondary flows in a pipe with a square cross section, swirling flow in a round pipe, paired vortex after bend of the pipe. The physical sense of more complex solutions of the wave equation has become clear relatively recently. Very similar structures were found in experimental studies using orthogonal decomposition (POD) of a turbulent pulsations field. This may mean that the eigenfunctions in the POD correspond to coherent structures that really arise in the flow. The results obtained confirm the hypothesis that secondary flows and coherent structures have a common nature. The solutions obtained in this paper can be used in processing the experiment as eigenfunctions for the orthogonal decomposition method. In addition, they can be used in direct numerical simulation (DNS) of turbulent flows

scholarly journals Generalizing the coupling between geometry and matter: $$f\left( R,L_m,T\right) $$ gravity

2021 ◽  
Vol 81 (7) ◽  
Author(s):  
Zahra Haghani ◽  
Tiberiu Harko
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Newtonian Limit ◽  
Gravity Models ◽  
Momentum Balance ◽  
Field Equations ◽  
Gravitational Fields ◽  
Generalized Poisson

AbstractWe generalize and unify the $$f\left( R,T\right) $$ f R , T and $$f\left( R,L_m\right) $$ f R , L m type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R, of the trace of the energy–momentum tensor T, and of the matter Lagrangian $$L_m$$ L m , so that $$ L_{grav}=f\left( R,L_m,T\right) $$ L grav = f R , L m , T . We obtain the gravitational field equations in the metric formalism, the equations of motion for test particles, and the energy and momentum balance equations, which follow from the covariant divergence of the energy–momentum tensor. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equations of motion is also investigated, and the expression of the extra acceleration is obtained for small velocities and weak gravitational fields. The generalized Poisson equation is also obtained in the Newtonian limit, and the Dolgov–Kawasaki instability is also investigated. The cosmological implications of the theory are investigated for a homogeneous, isotropic and flat Universe for two particular choices of the Lagrangian density $$f\left( R,L_m,T\right) $$ f R , L m , T of the gravitational field, with a multiplicative and additive algebraic structure in the matter couplings, respectively, and for two choices of the matter Lagrangian, by using both analytical and numerical methods.

scholarly journals Plane Symmetric Solutions of a Scalar-Tensor Theory of Gravitation

1977 ◽  
Vol 30 (1) ◽  
pp. 109 ◽  
Author(s):  
DRK Reddy
Keyword(s):  
General Relativity ◽  
Empty Space ◽  
Scalar Fields ◽  
Scalar Tensor Theory ◽  
Field Equations ◽  
Plane Symmetric ◽  
Zero Mass ◽  
Tensor Theory ◽  
Special Cases ◽  
Theory Of Gravitation

Plane symmetric solutions of a scalar-tensor theory proposed by Dunn have been obtained. These solutions are observed to be similar to the plane symmetric solutions of the field equations corresponding to zero mass scalar fields obtained by Patel. It is found that the empty space-times of general relativity discussed by Taub and by Bera are obtained as special cases.

On MBS Constraints and Projections

2021 ◽  
Author(s):  
Friedrich Pfeiffer
Keyword(s):  
Quadratic Form ◽  
Differential Equations ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Machine Process ◽  
Constraint Forces ◽  
Process Dynamics ◽  
Robot Tracking ◽  
Minimal Coordinates ◽  
Linear Differential

Abstract Constraints in multibody systems are usually treated by a Lagrange I - method resulting in equations of motion together with the constraint forces. Going from non-minimal coordinates to minimal ones opens the possibility to project the original equations directly to the minimal ones, thus eliminating the constraint forces. The necessary procedure is described, a general example of combined machine-process dynamics discussed and a specific example given. For a n-link robot tracking a path the equations of motion are projected onto this path resulting in quadratic form linear differential equations. They define the space of allowed motion, which is generated by a polygon-system.

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