Theoretical Basis of Rock Dynamics

We present another way to teach the least-action principle in classical mechanics. Beginning with Newton’s second law of motion, we keep finding equivalent propositions based mainly on the method of undetermined coefficients in a multivariable-dependent identity to arrive at the Euler–Lagrange equations and the conventional principle of stationary action $$\delta S=0$$ δ S = 0 with the boundary condition for the variation of the physical path. Pedagogically, by following the logical steps that we show, students can easily learn that Lagrangian mechanics naturally arises from Newtonian mechanics. This approach makes it further clear that Newton’s laws do not require the minimum action—true for a free particle, just as the shortest path between two points is a straight line. However, the action is not always the minimum once the potential energy is turned on. The derivation of the least-action principle from Newton’s second law may resemble Euclid’s approach to geometric axioms.

Dark matter could meet its nemesis on Earth

Introduction. One of the high points of elementary calculus is the derivation of Kepler's empirically deduced laws of planetary motion from Newton's Law of Gravity and his second law of motion. However, the standard treatment of the subject in calculus books is flawed for at least three reasons that I think are important. First, Newton's Laws are used to derive a differential equation for the displacement vector from the Sun to a planet; say the Earth. Then it is shown that the displacement vector lies in a plane, and if the base point is translated to the origin, the endpoint traces out an ellipse. This is said to confirm Kepler's first law, that the planets orbit the sun in an elliptical path, with the sun at one focus. However, the alert student may notice that the identical argument for the displacement vector in the opposite direction would show that the Sun orbits the Earth in an ellipse, which, it turns out, is very close to a circle with the Earth at the center. That would seem to provide aid and comfort to the Church's rejection of Galileo's claim that his heliocentric view had more validity than their geocentric one. Second, by placing the sun at the origin, the impression is given that either the sun is fixed, or else, that one may choose coordinates attached to a moving body, inertial or not. However, Newton's equations in their usual form hold precisely in inertial coordinates. Furthermore, it is an immediate consequence of Newton's Laws that the center of gravity of the two bodies can serve as the origin of an inertial coordinate system. One then finds that both the Earth and the Sun describe elliptical paths with a focus at the center of gravity of the pair. Third, by considering only the displacement vector, the treatment misses an opportunity to present one of the greatest recent triumphs of Newton's derivation of Kepler's Laws: the discovery of extrasolar planets during the past decade, none of which has been observed directly, but whose existence and orbital parameters have been deduced by detecting and analyzing the motion of the stars that they orbit. Two other considerations make the correct way of treating the subject particularly desirable. First, the current treatment for the displacement vector could be left as is, and only a little algebra is needed to show that the same vector referred to the center of gravity is a constant scalar multiple of the one referred to the sun. Second, it connects with an exciting area of current research in astronomy where the methods of elementary calculus have direct application. The purpose of this note is to spell out the slight modifications to the standard treatment of Kepler's Laws that are needed to obtain the additional and accurate information, and to provide a user's guide to the website http://exoplanets.org where a list of all the extrasolar planets discovered so far is maintained and constantly updated. The Basic Equations. Let r(t) be a (3-dimensional) vector function of the parameter t, and suppose that r(t) satisfies the differential equation

When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.

Space and time, inertia and gravity are unseen yet they are the most fundamental concepts needed to understand physics classically, relativistically, and quantumly. Non- Euclidean geometries and relativity theory have yielded new insights in treating the four interrelated metaphysical concepts. Minkowski linked time and space. Einstein linked Newton’s first and second laws of motion. Attempts to link electromagnetism / gauge forces to theories of gravity have been somewhat of a challenge. Could discrete inertia be the unrecognized link between quantum mechanics and relativity theory? Newtonian mechanics and Maxwell’s equations are based on the mathematics of continuous functions and relativistic effects count Newtonian laws as approximations. The classical period of physics, based on ideas of continuity came to an end when discreteness and quantum considerations, like the Planck black-body spectrum, became experimentally and theoretically relevant. General relativity and quantum mechanics have reshaped our way of thinking about classical geometric concepts of space and time as well as inertia and gravity. Unsolvable mathematical frameworks have not provided the inertial / gravitational and space / time superstructures needed to address the unification problem. The first attempts to link gravity to the behavior of subatomic particles originated in papers written by two Russian physicists. Sakharov proposed that Newtonian gravity could be interpreted as a dipolar van der Waals force that emerges from the action of quantum fluctuations of the empty vacuum. Zeldovich gave a new meaning to Einstein’s cosmological constant, as a measure of the curvature of space, allowing for new insights into cosmological inflation and dark energy. It will be argued in this paper that zero-point fluctuations which produce inertial reaction forces also produce the elasticity / curvature of spacetime suggested by Sakharov / Zeldovich. Is it possible that dynamic spacetime gives matter its curvature? We will attempt to show the feasibility of this idea by appealing to Rowlands quaternion representation of spacetime and gauge / gravity duality hypothesis. We will then argue that Sakharov’s induced continuous gravitational field emerges which balances the discrete inertial spacetime forces satisfying a zero-totality condition.

A. Compared to the discipline of international law, scholars of physics are blessed. While the principles of classical mechanics were theorized several centuries ago, quantum theory and the theory of relativity offer supplementary ways for describing how material objects and energy interact where classical mechanics does not provide an explanation. Thus, even in the absence of an all-comprising “world theory”, physicists have a wide array of workable theories at their service. By contrast, the “classical mechanics” of international law, i.e. the explanation of the most basic causal relationships between international legal norms and the behaviour of states as the main subjects of international law, are still subject to deep theoretical controversies. International legal doctrine presupposes that international law does have an impact and does not aim at questioning or further explaining this assumption. Traditional legal theories that see the essence of legal normativity in the possibility to trigger mechanisms of physical constraint often come to the conclusion that international law, in the absence of central enforcement mechanisms, is at best a primitive form of law. More recent enquiries into international legal theory from very different theoretical angles come to even less uplifting conclusions. Some argue that international legal norms are either entirely devoid of content because of their inherent indeterminacy and therefore prone to be captured by special interests. Others consider international law to be merely epiphenomenal because rational states would only consent to legal norms if, and as long as, they describe a behaviour they would choose anyway because it promises higher payoffs. In particular the latter critique put forward so forcefully by Jack Goldsmith and Eric Posner sent considerable shock waves through the invisible college of international lawyers. This is the background that needs to be kept in mind in order to assess the quantum of solace that Andrew Guzman’s new book provides.

Nonlinear drag force has been a research frontier in complex gas-solid systems. The literature has reported that the commonly-used drag correlations often overestimate drag force and, thus, cause unrealistic homogeneous flow structures in gas-solid fluidized beds of fine particles. For solving this problem, the structure-dependent drag model, derived from energy-minimization multi-scale approach, is used in discrete simulations of fluid catalytic cracking particles in a small riser. The gas phase is dealt with by computational fluid dynamics. Particles are considered as a discrete phase and described by Newton’s second law of motion. Gas-particle phases are coupled according to Newton’s third law of motion. Simulations show that use of structure-dependent drag model results in drag reduction, the effect of which is not so apparent as that in simulations of the two fluid model. The particle clustering tendency, however, is more distinct and leads to more heterogeneous flow structures in riser flow with a much greater amplitude of outlet solid flux fluctuations. Moreover, the behaviors of particle and gas back-mixing can be captured in the present simulations, which was supported by past simulations and experimental data. The simulation time resolution is discussed. The spring constant can be artificially brought down for safe setting of larger time step when modelling the collision process between fine particles with a higher calculation load. To appropriately mimic the continuous decay of van der Waals force may, however, need a much smaller time step. There is also an obvious effect of space resolution on simulations. When using a grid size smaller than 3 times the particle diameter, the simulated clusters turn extraordinarily large, and the effect of gas-solid back-mixing turns insignificant.

Derivation of an adiabatic time-dependent Hartree-Fock formalism from a variational principle

When drilling for oil and gas, maintaining wellbore stability is of primary importance to reduce non-productive time and trouble cost. Shale swelling causes many problems related to stability when drilling with water-based drilling fluids (WBDF). For many years, it remains the number one cause of time wastage and well abandonment. Different shale samples have different chemical compositions that affect their behavior when in contact with water. This factor makes laboratory-based analysis and characterization of the swelling mechanisms and action of swelling inhibitors extremely challenging. Moreover, the need to replicate different conditions at which clay–water interactions might occur necessitates using a different technique. Molecular dynamics (MD) simulation can be used as a supplement technique to help interpret experimental studies, test and improve a theoretical model, and provide empirical data in high-pressure and high-temperature condition of the borehole. MD simulation applies Newton’s second law of motion to describe particles’ movement in a classical system. The technique can be performed on the time scale of nanoseconds, and in three dimensions, it is thus sufficient for the study of clay–water interaction at a molecular level. It provides a unique view of the clay mineral interlayer and surface activities. This work reviews the progress in MD simulations of clay swelling and its inhibition mechanisms for application in petroleum drilling operations.

The equation F = ma, shorthand for Newtion's second law of motion, is the 1 + 1 = 2 of classical mechanics. The equation seems obvious and straightforward. It appears simply to translate an ordinary experience into measurable terms: push something and it moves.

We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.

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Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

Intramolecular (13)C kinetic isotope effects were determined for the dimerization of methacrolein. Trajectory studies accurately predict the isotope effects and support an origin in Newton's second law of motion, with no involvement of zero-point energy or transition state recrossing. Atomic motion reaction coordinate diagrams are introduced as a way to qualitatively understand the selectivity.

Bubble size distributions (BSDs) near a reference electrode (RE) in aqueous glycerol solutions of an electrolyte NaCl have been investigated under various gas superficial velocities (Us). BSD and voltage reading of the solution were measured by using a high-speed digital camera and a pH/voltage meter, respectively. The results show that bubble size (b) increases with liquid viscosity (μc) and Us. Self-similarity is seen and can be described by the log-normal form of the continuous number frequency distribution. The result shows that b controls the voltage reading in each solution. As b increases, the voltage increases because of gas bubbles interrupting their electrolyte paths in the solutions. An analysis of bubble rising velocity reveals that Stokes' Law should be used cautiously to describe the system. The fundamental equation for bubble formation was developed via Newton's second law of motion and shown to be the function of three dimensionless groupsWeber number, Bond number, and Capillary number. Af...