XLIII.—On the Gravitational Mass of a System of Particles

1949 ◽  
Vol 62 (4) ◽  
pp. 412-423
Author(s):  
G. L. Clark
Keyword(s):  
Gravitational Force ◽  
Line Element ◽  
Normal Component ◽  
Classical Mechanics ◽  
Physical Significance ◽  
Point Of View ◽  
Surface Integral ◽  
System Of Particles ◽  
Gauss Theorem ◽  
Relativistic Extension

SummaryIn classical mechanics the mass of a system of gravitating particles can be denned to be the mass of an equivalent particle which gives the same field at great distances, or alternatively the mass can be defined by means of Gauss' Theorem. Reference to the former procedure was made by Eddington and Clark (1938) in a discussion on the problem of n bodies. The relativistic extension of Gauss' Theorem has been investigated by Whittaker (1935) for a particular form of the line-element and for more general fields by Ruse (1935). The latter, treating the problem from a purely geometrical point of view, expressed the integral of the normal component of the gravitational force as the sum of two volume integrals. The physical significance of one of these integrals was quite obvious but the meaning of the other was far from clear. In this paper the terms in Ruse's result are examined as far as the order m2 in the case of a fundamental observer at rest and the 1938 discussion modified to bring the two investigations into line. It is concluded that the surface integral of the normal component of the gravitational force taken over an infinite sphere is –4π × the energy of the system.

scholarly journals Gauss' Theorem in a General Space-time

1935 ◽  
Vol 4 (3) ◽  
pp. 144-158 ◽  
Author(s):  
H. S. Ruse
Keyword(s):  
Gravitational Field ◽  
Total Mass ◽  
Gravitational Force ◽  
Line Element ◽  
Classical Mechanics ◽  
Closed Surface ◽  
General Space ◽  
Gauss Theorem ◽  
Image Position ◽  
Newtonian Constant

In classical mechanics Gauss' Theorem for a gravitational field states that, if S is a closed surface and N the component of gravitational force along the outward normal, thenwhere β is the Newtonian constant of gravitation and M is the total mass inside S. This result has recently been extended to general relativity by E. T. Whittaker,1 who, however, considered only the case of a statical gravitational field, the line-element of which is given by2where the coefficients U and αμν are independent of t. It is not immediately clear from his work whether the results are extensible to more general space-times.

General finite-volume framework for saddle-point problems of various physics

2021 ◽  
Vol 36 (6) ◽  
pp. 359-379
Author(s):  
Kirill M. Terekhov
Keyword(s):  
Saddle Point ◽  
Finite Volume ◽  
Navier Stokes ◽  
Saddle Point Problems ◽  
Surface Integral ◽  
Matrix Coefficients ◽  
Gauss Theorem ◽  
Stability Issue ◽  
Incompressible Elasticity ◽  
Vector Flux

Abstract This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

scholarly journals How Does Quantum Uncertainty Emerge from Deterministic Bohmian Mechanics?

2016 ◽  
Vol 15 (03) ◽  
pp. 1640010 ◽  
Author(s):  
A. Solé ◽  
X. Oriols ◽  
D. Marian ◽  
N. Zanghì
Keyword(s):  
Wave Function ◽  
Classical Mechanics ◽  
Bohmian Mechanics ◽  
The State ◽  
Quantum Uncertainty ◽  
Point Particles ◽  
Quantum Randomness ◽  
Quantum Dynamical ◽  
System Of Particles ◽  
Quantum Phenomena

Bohmian mechanics is a theory that provides a consistent explanation of quantum phenomena in terms of point particles whose motion is guided by the wave function. In this theory, the state of a system of particles is defined by the actual positions of the particles and the wave function of the system; and the state of the system evolves deterministically. Thus, the Bohmian state can be compared with the state in classical mechanics, which is given by the positions and momenta of all the particles, and which also evolves deterministically. However, while in classical mechanics it is usually taken for granted and considered unproblematic that the state is, at least in principle, measurable, this is not the case in Bohmian mechanics. Due to the linearity of the quantum dynamical laws, one essential component of the Bohmian state, the wave function, is not directly measurable. Moreover, it turns out that the measurement of the other component of the state — the positions of the particles — must be mediated by the wave function; a fact that in turn implies that the positions of the particles, though measurable, are constrained by absolute uncertainty. This is the key to understanding how Bohmian mechanics, despite being deterministic, can account for all quantum predictions, including quantum randomness and uncertainty.

scholarly journals Kinematic support modeling in Sage

2020 ◽  
Vol 28 (2) ◽  
pp. 141-153
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh ◽  
Sergei P. Karnilovich
Keyword(s):  
Rigid Body ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Seismic Isolation ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Vertical Plane ◽  
Base Plate ◽  
Euler Method ◽  
Point Of View

The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.

scholarly journals The Effect of Angular Momentum and Ostrogradsky-Gauss Theorem in the Equations of Mechanics

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Mathematical Models ◽  
Conservation Laws ◽  
Rarefied Gas ◽  
Classical Mechanics ◽  
Gauss Theorem ◽  
The Media ◽  
Velocity Circulation ◽  
The Times ◽  
The Many ◽  
Definition Of

There are many experimental facts that currently cannot be described theoretically. A possible reason is bad mathematical models and algorithms for calculation, despite the many works in this area of research. The aim of this work is to clarificate the mathematical models of describing for rarefied gas and continuous mechanics and to study the errors that arise when we describe a rarefied gas through distribution function. Writing physical values conservation laws via delta functions, the same classical definition of physical values are obtained as in classical mechanics. Usually the derivation of conservation laws is based using the Ostrogradsky-Gauss theorem for a fixed volume without moving. The theorem is a consequence of the application of the integration in parts at the spatial case. In reality, in mechanics and physics gas and liquid move and not only along a forward path, but also rotate. Discarding the out of integral term means ignoring the velocity circulation over the surface of the selected volume. When taking into account the motion of a gas, this term is difficult to introduce into the differential equation. Therefore, to account for all components of the motion, it is proposed to use an integral formulation. Next question is the role of the discreteness of the description of the medium in the kinetic theory and the interaction of the discreteness and "continuity" of the media. The question of the relationship between the discreteness of a medium and its description with the help of continuum mechanics arises due to the fact that the distances between molecules in a rarefied gas are finite, the times between collisions are finite, but on definition under calculating derivatives on time and space we deal with infinitely small values. We investigate it

ESTIMATING ORE MASSES IN GRAVITY PROSPECTING

Geophysics ◽  
1945 ◽  
Vol 10 (1) ◽  
pp. 50-62 ◽  
Author(s):  
Sigmund Hammer
Keyword(s):  
Potential Theory ◽  
Mass Distribution ◽  
Total Mass ◽  
Gravity Data ◽  
Point Of View ◽  
Theoretical Point ◽  
Chromite Ore ◽  
Ore Body ◽  
Gauss Theorem ◽  
Mass Estimate

The interpretation of the results of gravitational prospecting surveys is considered, from a theoretical point of view, in terms of the magnitude of the causative mass as distinct from the conventional interpretation in terms of the mass distribution (size, shape and depth). A general proof is given, based on Gauss’ Theorem in potential theory, that the former problem is unique and the uniqueness is illustrated by an analytical example which also serves to demonstrate the well‐known lack of uniqueness of the latter problem. Practical formulae are presented for estimating the total mass directly from the gravity data and the precision of the mass estimate is considered. The method is applied to a practical gravimeter survey over a known chromite ore body and the estimated mass is found to be in excellent agreement with estimates from core drilling.

Rotation as a Source of Accelerating Expansion of the Universe

2021 ◽  
Author(s):  
Andrzej Szummer
Keyword(s):  
Gravitational Force ◽  
Cosmological Parameters ◽  
Classical Mechanics ◽  
Astronomical Observations ◽  
Accelerating Expansion ◽  
Expansion Of The Universe ◽  
The Difference ◽  
Changes Over Time ◽  
The Universe ◽  
Over Time

Abstract Assuming a hypothesis, that the universe is rotating from the very beginning – as soon as it appeared- creates new possibilities to explain accelerating expansion of the universe. A spinning universe is under the action of two enormous forces: gravitational force and centrifugal force. The difference between the two forces has been shown to give the resultant force that causes the expansion of the universe to accelerate. Applying classical mechanics as a method, I calculated the magnitude of this acceleration, the time when it appeared and how it changes over time. By applying only recognized cosmological parameters, interesting results were obtained that can be checked with astronomical observations. The presence of acceleration of expansion causes the rate of expansion of the universe to continue to increase, which is consistent with astronomical observations. However, the speed of this increase in the rate of expansion becomes slower over time.

The PI Decision Theory Application of Multi-Round Profile Control

2013 ◽  
Vol 295-298 ◽  
pp. 3302-3305
Author(s):  
Zuo Chen Li
Keyword(s):  
Decision Theory ◽  
Physical Significance ◽  
Point Of View ◽  
Oil Field ◽  
Theoretical Point ◽  
Technology Research ◽  
Research Directions ◽  
Technology Application ◽  
Profile Control ◽  
Theory Application

In this paper, from a theoretical point of view, introduce systematically the PI decision index and its physical significance, discuss in relation to technology application and method in multi-round Profile control, and propose some adjustments in the grape flower oil field applications of the technology research directions.

Geometric Hamiltonian quantum mechanics and applications

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630017 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Star Product ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Point Of View ◽  
Quantum Theories ◽  
Geometric Point ◽  
Set Up

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

Sign in / Sign up

Export Citation Format

Share Document