Newtonian Form Research Articles

A stochastic derivation of the Sivashinsky equation for the self-turbulent motion of a free particle

Within the framework of the Kershaw approach and of a hypothesis on spatial stochasticity, the relativistic equations of Lehr and Park, Guerra and Ruggiero, and Vigier for stochastic Nelson mechanics are obtained. In our model there is another set of equations of the hydrodynamical type for the drift velocityv i(x j,t) and stochastic velocityu i(x j,t) of a particle. Taking into account quadratic terms in l, the universal length, we obtain from these equations the Sivashinsky equations forv i(x j,t) in the caseu i →0. In the limit l →0, these equations acquire the Newtonian form.

Gravitational two-body problem—A source theory viewpoint

An analysis of the semirelativistic gravitational two-body problem based on Schwinger's source theory is given. Our treatment is purely classical but nongeometrical. Only the large distance behavior of the gravitational stress tensor is seen to be relevant for order G 2 contributions. For a gravitational stress tensor that has a pure Newtonian form, only the traceless choice is seen to be consistent with Einstein's theory. We find results in agreement with the earlier results of Einstein, Infeld, and Hoffmann, and Barker and O'Connell for spin-2 graviton exchanges between Dirac particles, and with Börner, Ehlers, and Rudolph for spin precession in theories with arbitrary post-Newtonian parameter γ. The source approach clearly displays the analogy between gravitational interactions and classical electrodynamics. We also discuss the general relationship between the periastron advance and spin-precession frequencies for a class of gravitation theories. Brief estimates of the various spin-precession effects are given for the Hulse-Taylor pulsar.

Turbulent Flow of Power-Law Fluids

Velocity profiles for the steady incompressible turbulent flow of power-law fluids through circular conduits were calculated by numerical quadrature considering both the viscous stresses and the Reynolds' stresses operative over the entire cross section of the flow. A non-Newtonian stress coeffiecient, K sub Sn, was introduced along with mixing length, l, so that the Reynolds' stresses could be put in a form similar to that of the viscous stresses. The form of the Reynolds' stresses thus obtained reduced to the Newtonian form when the flow behavior index, n, equaled unity. Both l and K sub Sn were assumed dependent upon the Reynolds number. Similarly, friction factors for the turbulent flow of power-law fluids were calculated by numerical quadrature. The calculated friction factors were difference between calculated and measured friction factors Reynolds numbers from 2,670 to 108,800. The mean absolute difference between colculated and measured friction factors for this sample was 3 percent with a standard deviation of 1.9 percent.

Dynamical symmetries and constants of the motion for classical particle systems

By formulating the conditions for dynamical symmetry mappings directly at the level of the dynamical equations (which are taken in the form of Newton's equations, Lagrange's equations, Hamilton's equations, or Hamilton-Jacobi equation), we derive new expressions for dynamical symmetries and associated constants of the motion for classical particle dynamical systems. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (a) x̄i =xi+δxi [δxi≡ξi(x)δa] with associated changes in the independent variable t (path parameter) defined by (b) δt≡{∫2φ[x(t)]dt+c} δa. A generalized form of the related integral theorem (a method for obtaining constants of the motion based upon deformations of a known constant of the motion under dynamical symmetry mappings) is obtained. We take the ``Newtonian form'' of the dynamical equations to have a coordinate-covariant structure with forces defined by a general polynomial in the velocities and obtain dynamical symmetry conditions for all such systems. For the special case of conservative systems the related integral theorem is applied. Based upon Lagrange's equations with L=L(xi,ẋi) we find the conditions for dynamical symmetry mappings may be expressed in the form (c) (∂/∂xj)[δL+L(d/dt)(δt)]−(d/dt)(∂/∂ẋj)[δL+L(d/dt)(δt)]=−2φ,j[(∂L/∂ẋi)ẋi−L]δa. From this form we obtain a new formula for concomitant constants of the motion: (d) [∂(δL)/∂ẋj] ẋj −δL = k. By use of the related integral theorem such constants of the motion can be expressed as deformations of the energy integral under the dynamical symmetry mappings defined by (c). A short derivation of the Noether identity is given which is independent of the integration processes of Hamilton's variational principle. For mappings of the type (a), (b) ``Noether type'' symmetries and associated constants of the motion are formulated. For a conservative dynamical system with L≡(1/2)gijẋiẋj − V(x) we find such Noether symmetries are basically conformal motions, while those derived from (c) are basically projective collineations. For such systems the constants of the motion (d) are evaluated and shown in general to differ from those obtained from the Noether method. We show for conservative dynamical systems that the formulation of dynamical symmetry mappings directly at the level of the Hamilton-Jacobi equation leads to the Noether symmetry conditions. Dynamical symmetry conditions are formulated for Hamilton's equation in phase space and shown to be more general than canonical transformations. The formulation of the related integral theorem in phase space is found to be a generalization of Poisson's theorem. For systems with H(xA), A = 1,…,2n, it is an immediate observation that δH induced by a symmetry mapping is a constant of the motion. Application to the isotropic harmonic oscillator shows both symmetric tensor and angular momenta constants of the motion are obtained in this manner. An additional constant of the motion ∂A ξA−2φ(xA) is shown in general to be a concomitant of a phase space symmetry transformation.

On the maintenance of the axisymmetric part of the flow in the atmosphere

The maintenance of the axisymmetric component of the flow in the atmosphere is investigated by means of a steady-state, quasi-geostrophic formulation of the meteorological equations. It is shown that the meridional variations in the time-averaged axisymmetric variables can be expressed as the sum of three contributions, one being due to the eddy heat transport, another to the eddy momentum transport, and a third to the convective-radiative equilibrium temperature which enters the problem through the specification of a Newtonian form of diabatic heating. The contributions by the large scale eddies are evaluated through the use of observed values for the eddy heat and momentum transports. The contributions from each of the three forcing mechanisms to the temperature and zonal wind fields are invstigated individually and found to be of about equal importance. The sum of the three contributions are also presented for the temperature, the zonal wind, the stream function associated with the mean meridional circulation and the corresponding vertical motion. Although the results fail to reproduce the main observed features of the lower stratosphere, they are found to be in good agreement with observations in the middle latitude troposphere. At any pressure level, for example, the computed mean zonal wind has a jet-like profile and the axis of the jet is found to slope to the south with height, as observed in the atmosphere.

The stress system in a suspension of force-free particles

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

A theoretical study of the annual variation of atmospheric energy

The zonally averaged equation for a two-level, quasi-geostrophic model are used to calculate the annual mean values and the first harmonic of the annual variation of the midtropospheric temperature and the zonal winds at the upper and lower levels. The diabatic heating in the model is of a simple Newtonian form. The effects of surface skin friction and of internal friction are included together with a simple parameterization of the meridional eddy transport of sensible heat, while the transport of zonal relative momentum by the eddies is neglected, thereby restricting the motion to the largest meridional scale. From the solution of the equations it is possible to calculate the annual mean values and the first harmonic of the annual variation of the amounts of zonal available potential energy and zonal kinetic energy and, also, the generation of zonal available potential energy, the convertions from this form of energy to eddy available potential energy and to zonal kinetic energy, and the dissipation of zonal kinetic energy by friction. The results show that the simplified model is capable of reproducing several important aspects of the annual variation of atmospheric energetics. The phase relationships involving the time lag between the generation of zonal available potential energy and the dissipation of kinetic energy as well as the time lag between the seasonal forcing and the maxima of zonal available potential energy and zonal kinetic energy are well reproduced. There are differences between the theoretical results and observational studies with respect to the intensity of the general circulation and the amounts of energy. The sensitivity of the theoretical solution to the numerical values of the physical parameters is investigated. DOI: 10.1111/j.2153-3490.1970.tb01931.x

An improved theory of gravitation. Part I

A theory of gravitation is developed from assumptions that differ as little as possible from those of special relativity and the Newtonian theory of gravitation. As in special relativity, one assumes the existence of preferred coordinate systems (Newtonian charts) in which the nondiagonal components of the metric vanish, and in which the spatial, diagonal components are equal. The metric is determined by a single real function, the gravitational potential, which is assumed, as in the Newtonian theory, to be arbitrary to the extent of an additive constant. A uniqueness theorem is proved for Newtonian charts, and the functional dependence of the metric on the gravitational potential is determined (apart from two constants, which are later fixed by requiring that the equations of motion of a particle have the correct, nonrelativistic limit, and that the potential due to a fixed particle have the Newtonian form at great distances). By a simple change in the units of space and time, the geometry is made Minkowskian. A similar change in the units of mass makes the theory formally similar to special relativity. Particle dynamics is developed. The red shift and the deflection of light by a star are calculated, and agree with the Einstein results. The combination of the assumptions that the potentials due to particles are additive and that the potential due to a fixed particle is not proportional to 1/r, is shown to lead to difficulties. The weight of a simple system is found to be proportional to its total energy, including its gravitational interaction energy. Continuous, static mass distributions are considered. A field equation is derived for the static gravitational potential, and an expression for the energy density of the static gravitational field. The field equation is modified by assuming that the gravitational energy density is itself a source of the gravitational potential. The potential due to a static, spherically symmetric body is calculated, and the perihelion advance of a planet is found to be 11/12 of the Einstein value, in good agreement with the results of Dicke.

Scattering of Massless Scalar Waves by a Schwarzschild ``Singularity''

This paper investigates the scattering and absorption of scalar waves satisfying the equation φ;μ;μ=0 in the Schwarzschild metric. This problem has been previously considered by Hildreth. We find, for a Schwarzschild mass m, the following cross sections in the zero-frequency limit for s-waves: σ(absorption) = 0, dσ/dΩ ≃ [c + ⅓(2m) ln (2mω)]2, where c is a constant of order m. These results disagree with the previous calculation. We exhibit a method of solution for the equation. Its limiting (Newtonian) form, with suitable identification of the coefficients, is the problem of Coulomb scattering in non-relativistic quantum mechanics. By demanding coordinate conditions which for large l allow the usual Coulomb results in a partial-wave expansion, we are able to define a partial-wave cross section. The (summed) differential cross section for small frequencies inherits the logarithmic behavior of the s-wave part, which is the only contribution explicitly calculated. (The l ≠ 0 contributions and the behavior of the cross sections for ω ≠ 0 are qualitatively indicated.) Cosmological considerations are given which cut off this divergence.

A Variational Principle for the Mesh-Type Analysis of a Mechanical System

Abstract The mesh-type equations for a dynamical system consisting of N-mass points are derived from the Newtonian form of the equations of motion. The appropriate Lagrangian for a variational principle is established for these equations.

Six-color photometry at the Lick Observatory.

view Abstract Citations (1) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Six-color photometry at the Lick Observatory. Stebbins, Joel ; Kron, Gerald E. ; Smith, J. Lynn Abstract The Crossley reflector has been adapted to the Newtonian form, and a photoelectric photometer with interchangeable receivers has been installed. Improved circuits and a Brown recorder have been substituted for the previous arrangement with a galvanometer. Six-color measures in the range from 3530 to 10,300 A have been completed for 125 stars additional to the 238 stars previously published by Stebbins and Whitford. For determination of the scale of stellar temperatures the stars have been compared with a standard lamp run at a color temperature of about 25000 absolute as calibrated by the U. S. Bureau of Standards. This lamp with a small hole in front of the tungsten ribbon filament serves as an artificial star which, operated by portable batteries, has been carried in a jeep to different peaks of Mount Hamilton at distances of 1100 and 3200 feet from the telescope. At the shorter distance this star appears of about zero visual magnitude, which is cut down five magnitudes by a wire gauze screen; at the longer distance a 2.5-mag. screen equalizes the intensity. Thus the real stars and the artificial star are observed under the same optical conditions, and the corrections for atmospheric extinction have been satisfactorily determined. The horizontal extinction at 1100 feet on Mount Hamilton is practically nil for wave lengths 4200 A and longer. The final values of stellar temperatures will depend upon the calibration of the photocell and filters now being completed in the laboratory. Other results include a new group of light- curves for the variable star, TI Aquilae. The so- called "hump" or pause in the continuous varlation of the star is present in all colors. In the period of seven days the range is 1.5 mag. in the ultraviolet and 0.4 mag. in the infra-red. The same retardation of phase for longer wave lengths is found as for a Cephei. Six-color curves for Algol are also nearly complete. Lick Observatory, Mount Hamilton, Calf. Publication: The Astronomical Journal Pub Date: April 1950 DOI: 10.1086/106352 Bibcode: 1950AJ.....55...80S full text sources ADS |

Kinematics, dynamics and the scale of time-II

1—In earlier investigations (Milne 1932, 1933, 1935, 1936 a, b ) the author has developed certain kinematic representations of the expanding universe. Amongst such representations is the substratum , or "smoothed-out" universe. Without appeal to the empirical laws of dynamics and gravitation, or to "field equations" or to assumed properties of geodesics in space-time, it was shown how to obtain the equation of motion of a free test-particle in the presence of a substratum, purely from the definition of a substratum. This equation of motion was then used to construct the dynamics of a particle under external force. The various dynamical quantities—mass, momentum , force, worh, energy—arose naturally from the study of a certain identity, and ttre relations between them were in many eases similar to those in the Newton-Einstein mechanics. But the equation of motion itself was formally different from the Newton-Einstein empirical equation of motion; it contained a term which could lie taken to represent the gravitational effect of the substratum on the particle; this term was Used to calculate the universal value of the "constant" of gravitation. In Part I (1937) of the present communication it was shown that in the clocks supposed to be carried by the particle-observers of the substratum are re-graduated from "kinematic" time t to a certain "dynamical" time T , and all derived measures adjusted correspondingly, then the characteristic additional term in the equation of motion disappeared and the equation of motion assumed, at least locally and for small velocities, a strictly Newtonian form. The relation between t and T was T = t 0 log ( t / t 0 ) + t 0 . (1)

On the quantization of a theory arising from a variational principle for multiple integrals with application to born's electrodynamics

The transition from classical mechanics to quantum mechanics has been formulated in various ways. But all these ways have the common feature that they use as starting point mechanics not in its Newtonian form, but in the form which it was given by Lagrange and Hamilton. The essence of this form consists in the fact that the differential equations are represented as Euler equations of a variational principle for v functions, the generalized coordinates, of one variable, the time. In its purely mathematical aspect the quantization method may, therefore, be regarded as a mode of taking certain notions and certain relations arising from a variational principle for several dependent and one independent variable and of giving them a new meaning.

A Study of Relative Motion in Connection with Classical Mechanics

Derivation of equations of motion of the second order, containing relative coordinates only.---This article is a continuation of the work of many authors, among others Berkeley, Neumann, Maxwell, Lange, Mach, Boltzmann, F\"oppl, and Kolkmeyer, whose work is briefly discussed. The main difficulty in Newton's mechanics seems to be to define the system of axes (inertial system) to which the motion is referred. Three solutions have been proposed. F\"oppl defined the system on a relative basis. Another and apparently more direct solution is to refer the motion of particles to other particles, that is, to use "relative coordinates" in the equations of motion. A third solution would be to obtain invariant equations, the same for all systems of reference. Introducing an hypothesis stated by F\"oppl, that for a system with its origin at the center of mass the angular momentum of the universe vanishes, invariant equations are obtained from the Newtonian, (though this is sometimes denied to be possible), and they degenerate into the Newtonian form if certain functions of coordinates and velocities occurring in them become zero by a suitable choice of the system of reference. It is believed, however, that such equations do not offer a complete solution of the problem of relativation of motion so as to satisfy the physicist; whereas the introduction of relative coordinates does. In this article this second solution has been applied to the Newtonian theory. Equations of motion are derived by introducing F\"oppl's hypothesis, which are of the second order and contain relative coordinates only. These equations may be taken as the basis for a complete system of mechanics. As examples the systems of Ptolemy and of Kepler are worked out.Application of relative coordinates to Einstein's theory of gravitation.---It is believed that the possibility of introducing relative coordinates is not limited to the Newtonian theory. As an example of the application of these coordinates to Einstein's theories, the case of the motion of a heavy and a light particle is considered and it is suggested that an hypothesis similar to F\"oppl's may give equations of motion into which only the distance apart of the particles measured by an observer on the heavier, enters.

II. On a cassegrain reflector with corrected field

The great advantage enjoyed by the reflecting telescope is its equal treatment of rays of all colours, and the geometrical defects or aberrations of its field are less than those of many of the older refractors. The most serious of these defects is coma, owing to which different zones of the objective do not place the light which they receive from the same object-point symmetrically around any common centre in the image area, but arrange it in a radial fan or flare, the light from the outer zones being most diffused; besides spoiling the image this tends to neutralize, for any except narrow fields, the value of extended aperture in the objective as a light-collector. In the refractor this can be and is now always met by adjusting the curves of the two lenses, for when achromatism, as far as possible, and spherical aberration are allowed for, there still remains one unused datum; in old forms this was often used to make the inner curves contact curves that might be cemented together if it was convenient to do so, but it is properly employed to extinguish coma. But with the reflector the case is different. In the Newtonian form there is only one available surface, and when this is made a paraboloid to cure spherical aberration, nothing is left to adjust. In the Gregorian or Cassegrain forms there are two curved surfaces and, theoretically, these would offer means to correct two faults. An illuminating study of the possibilities of a system of two mirrors has been made by Schwarzschild in his ‘Untersuchungen zur Geometrischen O ptik’; I shall deal with its outcome below. Its general tenor is comprehensive and exploratory rather than detailed, and it remains doubtful whether any of the forms which he indicates for the reflector, at the point at which his research stops, could actually be made successfully upon a scale that would show their advantages. My own purpose in the present paper is essentially a practical one. I have in mind throughout a telescope of large aperture and considerable focal length, and seek to devise a correction for the faults of its field which shall leave its achromatism unimpaired, which can really be made and which shall effect its purpose without employing any curves and angles outside those that are already known to work well. It has been said that “an object-glass cannot be made on paper,’’ but the possibilities of new and somewhat complicated constructions must in all cases first be demonstrated on paper, since practice can never conveniently vary more than a single factor at a time. Study is directed to the Cassegrain because of the great advantage which this design possesses in shortening the tube of the instrument for given focal length, and in placing the observer at the lower, in place of at the upper, end of it. The best introduction to the subsequent work will be in the form of a few remarks upon Schwarzschild’s results. These are not meant as a complete criticism or estimation of it but are merely such as arise naturally in relation to the points with which I deal afterwards. The traditional form of Cassegrain telescope consists of a great concave mirror faced by a small convex one, which is placed between the great mirror and its principal focus, and throws the image out through a hole cut centrally in the great mirror. The small mirror increases the effective focal length in the ratio of its distances respectively from the final principal focus and from the principal focus of the great mirror.

The Existence of Absolute Motion

IN discussing this question it is surely necessary to place stress on the contrast between the places of absolute direction and absolute position in dynamics. The result of observation is that the laws of motion are competent to explain such phenomena as nutation and retain the simple Newtonian form when certain directions which can be found with reasonable accuracy are assumed to be absolute. The contrary assumption, that these directions were not absolute, but moving with absolute angular velocities, say of the order of one degree per second, would necessitate a re-statement of the laws of motion involving great loss of simplicity. In the same way, we cannot without loss of simplicity suppose that the acceleration of the earth with respect to the centre of the solar system differs greatly from the absolute acceleration, and suggest that the material universe has an absolute acceleration of the order of one hundred miles per second per second.

XXXIII. Account of some observations tending to investigate the construction of the heavens.

In a former paper I mentioned, that a more powerful instrument was preparing for continuing my reviews of the heavens. The telescope I have lately completed, though far inferior in size to the one I had undertaken to construct when that paper was written, is of the Newtonian form, the object speculum being of 20 feet focal length, and its aperture 18 7/10 inches.