Total Linear Momentum Research Articles

A simple matrix method for the force balancing of planar multi-degree-of-freedom linkages

The shaking or fluctuating reaction forces produced by the moving links of a mechanism negatively impact its adequate performance due to the inherent generation of base vibrations, fatigue stresses, accuracy loss of the end-effector or moving platform, and so on. To increase the functionality and durability of the mechanism is therefore strongly advised to eliminate these parasitic forces. This article provides the balancing conditions for planar multi-degree-of-freedom linkages by canceling the total shaking force generated by the moving links of the mechanism. The method avoids the classical computation of the total linear momentum employed in existing methods. The balancing of three representative linkages proves the reliability of the proposed method.

Practical phase-space electronic Hamiltonians for abinitio dynamics.

Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ') and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.

Surface hopping, electron translation factors, electron rotation factors, momentum conservation, and size consistency.

For a system without spin-orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e., the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors) and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass [which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors below, cf. Eq. (96)]. The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing the new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum.

Radiation-reaction and angular momentum loss at the second post-Minkowskian order

We compute the variation of the Fokker-Wheeler-Feynman total linear and angular momentum of a gravitationally interacting binary system under the second post-Minkowskian retarded dynamics. The resulting $O({G}^{2})$ equations-of-motion-based, total change in the system's angular momentum is found to agree with existing computations that assumed balance with angular momentum fluxes in the radiation zone.

Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows

AbstractIn this paper, we present consistent and inconsistent discontinuous Galerkin (dG) methods for incompressible Euler and Navier–Stokes equations with the kinematic pressure, Bernoulli function and EMAC function. Semi‐ and fully discrete energy stability of the proposed dG methods are proved in a unified fashion. Conservation of total energy, linear, and angular momentum is discussed with both central and upwind fluxes. Numerical experiments are presented to demonstrate our findings and compare our schemes with conventional schemes in the literature in both unsteady and steady problems. Numerical results show that global conservation of the physical quantities may not be enough to demonstrate the performance of the schemes, and our schemes are competitive and able to capture essential physical features in several benchmark problems.

Time-dependent forces between a swift electron and a small nanoparticle within the dipole approximation

In this paper we calculate the time-dependent forces between a swift electron traveling at constant velocity and a metallic nanoparticle made of either aluminum or gold. We consider that the nanoparticle responds as an electric point dipole and we use classical electrodynamics to calculate the force on both the nanoparticle and the electron. The values for the velocity of the electron and the radius of the nanoparticle were chosen in accordance with electron microscopy observations, and the impact parameter was selected to fulfill the constraints imposed by the dipole approximation. We found that there are times when the force on the nanoparticle is attractive and others when it is repulsive, and show that this is due to the delayed electromagnetic response of the nanoparticle. To establish the limits of validity of our approach, we calculate the total linear momentum transfer to the nanoparticle, and compare it with results obtained, in frequency space, using the full multipole expansion of the fields induced on the nanoparticle, considering the effects of electromagnetic radiation.

Reducing orbital eccentricity in initial data of black hole–neutron star binaries in the puncture framework

We develop a method to compute low-eccentricity initial data of black hole--neutron star binaries in the puncture framework extending previous work on other types of compact binaries. In addition to adjusting the orbital angular velocity of the binary, the approaching velocity of a neutron star is incorporated by modifying the helical Killing vector used to derive equations of the hydrostationary equilibrium. The approaching velocity of the black hole is then induced by requiring the vanishing of the total linear momentum of the system, differently from the case of binary black holes in the puncture framework where the linear momentum of each black hole is specified explicitly. We successfully reduce the orbital eccentricity to <~0.001 by modifying the parameters iteratively using simulations of ~3 orbits both for nonprecessing and precessing configurations. We find that empirical formulas for binary black holes derived in the excision framework do not reduce the orbital eccentricity to ~0.001 for black hole--neutron star binaries in the puncture framework, although they work for binary neutron stars.

Galilean covariance of quantum-classical hybrid systems of the Sudarshan type

We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kind of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.

Integrating dissipative particle dynamics with energy conservation.

To suppress secular energy drift in dissipative particle dynamics simulations with energy conservation, we introduce an additional pairwise particle dynamics that allows for a microscopic energy fluctuation while conserving the total linear momentum of the system exactly. The pairwise dynamics may be regarded as an adaptation of the thermostat in isothermal dissipative particle dynamics, but leads to microcanonical instead of canonical ensemble at equilibrium and allows for a nonuniform temperature field at a steady state. The method is also effective in suppressing secular energy drift when used in combination with reverse nonequilibrium molecular dynamics moves. All of the equilibrium and transport properties we computed were unaffected by this additional pairwise dynamics.

Impact analysis of PEEK/ceramic/gelatin composite for finding behind the armor trauma

A ceramic-based armor system must arrest bullet penetration and dissipate a large amount of impact energy. Taking the total linear momentum transferred to the wearer of a ceramic armor as a measure of the behind-armor ballistic/blunt trauma (BABT), we study the effect on the BABT of adding a thin polyether-ether-ketone (PEEK) layer on the face of a Silicon Carbide (SiC) ceramic target impacted at normal incidence by a full metal jacket bullet traveling at a low velocity. A human torso is simulated by a block of ballistic gelatin. Three-dimensional non-linear large deformations of the system are numerically analyzed using the commercial finite element software ABAQUS/Explicit with the full integration rule to evaluate elemental matrices. Sensitivity tests are conducted with random samples generated by the Latin Hypercube Sampling technique to delineate the influence of the static yield stress, strain hardening, Young’s modulus and the mass density of the PEEK layer as well as of the shear modulus, the mass density and the maximum hydrostatic tensile stress of the ceramic on the BABT. It is found that reducing the acoustic mismatch between the PEEK and the ceramic increased the plastic energy dissipated in the ceramic. For normal impact at 180 m/s of a bullet having kinetic energy of 130 J, the 2 mm thick surface layer decreased the total peak normal force, the normal impulse transmitted, and the maximum pressure transferred from the SiC target to the gelatin layer by 19%, 18% and 32%, respectively. Simulations using a reduced integration rule significantly under-predict the final kinetic energy and the back face displacement of the bullet as compared to those using the full integration rule. For the 2 mm thick PEEK layer located on the top, the middle and the bottom of the SiC plate, the back face displacement is the least for the layer located in the middle but the BABT increased.

On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles

We prove the existence of uncountably-many physical scattering maps for non-spherical hard particles which, when used to construct global-in-time weak solutions of Newton’s equations of motion, conserve the total linear momentum, angular momentum and kinetic energy of the particle system for all time. We prove this result by first exhibiting the non-uniqueness of a classical solution to a constrained Monge–Ampere equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we briefly discuss the relevance of our observations to the statistical mechanics of hard particle systems.

Momentum transfer in hypervelocity cratering of meteorites and meteorite analogs: Implications for orbital evolution and kinetic impact deflection of asteroids

Kinetic impact is regarded as an effective way to divert an asteroid on a collision course with the Earth. However, asteroids show a wide diversity in their mineralogy, porosity, and water content. Asteroids range in composition from primitive undifferentiated objects to differentiated objects. Among the undifferentiated asteroids they range from more reflective, carbon-poor bodies to darker, carbonaceous bodies. Porosities of asteroids range from 0 to >50%, with most >20%, and some asteroids exhibit a 0.7 µm water feature in their reflection spectra. Mineralogy, porosity and hydration are each expected to influence the momentum transferred in hypervelocity collisions. We conducted a series of measurements of the post-impact momentum, which is characterized by a factor β, the ratio of the total linear momentum acquired by the target to the momentum of the impactor. We measured β for anhydrous meteorites, which sample their asteroidal parent bodies, spanning a wide range of porosities: including 7 samples of the CV3 carbonaceous chondrite Northwest Africa (NWA) 4502 (2.1% porosity), 7 samples of the ordinary chondrite NWA 869 (6.4% porosity), 4 samples of the ordinary chondrite Saratov (15.6% porosity), and, to extend our measurements to higher porosity than is found among meteorites, 2 samples of terrestrial pumice (80% porosity). We also measured β for hydrous meteorite analog targets: including 2 samples of terrestrial serpentine (17.9% porosity) and 4 samples of terrestrial montmorillonite (51.5% porosity), the two clay minerals that dominate the composition of the hydrous CI carbonaceous chondrite meteorites, as well as 4 samples of hydrous meteorite analog material prepared by powdering and hydrating an anhydrous carbonaceous chondrite. We found that for both the anhydrous and the hydrous samples β decreased with increasing porosity, consistent with hydrocode modeling. However, the β values we measured for the ∼5 km/s impacts onto these anhydrous samples, with β = 3.55 for NWA 4502, 2.69 for NWA 869, 2.10 for Saratov, and 2.15 for pumice, are larger than results from hydrocode modeling for 10 km/s impacts into relatively strong, porous rock targets. The β values for the moderate porosity (17.9%) hydrous serpentine targets (β = 4.70), the highly porous (51.55%) hydrous montmorillonite targets (β = 2.79), and the moderately porous CI-analog targets (β = 2.99) are each significantly larger than the β value for anhydrous targets of comparable porosities. This is likely due to jetting of water vapor, which could significantly affect the deflection of hydrous asteroids and icy comets in natural or human-induced collisions.

Duality in a hyperbolic interaction model integrable even in a strong confinement: multi-soliton solutions and field theory

Models that remain integrable even in confining potentials are extremely rare and almost non-existent. Introducing an external potential in integrable models, breaks the conservation of even simple quantities such as total linear momentum, and hence, the system no longer remains integrable in general. The Calegero family provides a unique and rare opportunity to study models, integrable even in external potentials. Here, we consider a one-dimensional hyperbolic interaction model, which we call as the hyperbolic Calogero (HC) model. This is classically integrable even in a confining potential (which have box-like shapes). We present a first-order formulation of the HC model in an external confining potential. Using the rich property of duality, we find multi-soliton solutions of this confined integrable model. Absence of solitons correspond to the equilibrium solution of the model. We demonstrate the dynamics of multi-soliton solutions via brute-force numerical simulations. We study the physics of soliton collisions and quenches using numerical simulations. We examine the motion of dual complex variables and find an analytic expression for the time period in a certain limit. We give the field theory description of this model and find the background solution (absence of solitons) analytically in the large-N limit (where N is the number of particles). Analytical expressions of soliton solutions are obtained in the absence of external confining potential. Our work is of importance to understand the general features of trapped interacting particles that remain classically integrable and can be of relevance to the collective behaviour of trapped cold atomic gases as well.

Conservação da velocidade do CM de duas partículas sob a ação de forças de atrito em movimento unidimensional (D = 1)

A força de atrito está presente no cotidiano dos alunos e a compreensão de sua ação sobre os corpos aproxima a Ciência de sua experiência. A lei de conservação do momento linear total, envolvendo sistemas de partículas sob a ação de forças dissipativas externas, não é geralmente discutida nos livros-texto de Mecânica Newtoniana utilizados nos ensinos médio e superior, nas instituições de ensino brasileiras. Discutimos neste trabalho o movimento mais simples que duas partículas podem ter: o movimento unidimensional em sentidos opostos, sob a ação de forças de atrito cinético externas. Mostramos que a conservação do momento linear do conjunto formado por essas partículas, sob a ação de duas forças de atrito cinético, ocorre apenas em partes do movimento unidimensional e quando elas se deslocam em sentidos opostos. Aproveitando o grande envolvimento dos alunos com aplicativos em celulares, apresentamos uma montagem experimental que verifica a conservação da velocidade do centro de massa (CM) desses dois corpos. O aparato experimental é simples e de baixo custo, podendo ser facilmente reproduzido em sala de aula como estratégia para estimular os estudantes a trabalharem com ferramentas matemáticas e métodos experimentais, de forma a discutirem situações físicas que incluem as forças de atritos presentes no seu dia a dia.

Generalized self-consistent reservoir model for normal and anomalous heat transport in quantum harmonic chains.

Fictitious stochastic reservoirs incorporate scattering and dephasing mechanisms into the system in contact with these reservoirs. The reservoir-system coupling is described by the related self-energy in terms of the nonequilibrium Green's function formalism or equivalently the quantum Langevin equation formalism. In this study, we investigate thermal transport in a finite segment of an infinitely extended quantum harmonic chain with an equal self-energy at each site by using the self-consistent reservoir approach. In this setup, the entire system is lattice translation invariant so that mismatched boundaries are excluded from the model. Solving the Landauer-Büttiker equations under the self-consistent adiabatic condition, we quantitatively elucidate a thermally induced crossover of ballistic-to-diffusive transport and its scaling relation prescribed by a temperature-dependent mean free path. It is also shown that normal transport emerges in the diffusive limit for a linear self-energy, while nonlinear higher-order ones generically lead to anomalous transport. Physical implications of these observations are discussed in terms of the persistence of a massless Goldstone mode as well as the conservation of total linear momentum.

Revisiting the peridynamic motion equation due to characterization of boundary conditions

The peridynamic motion equation was investigated once again. The origin of incompatibility between boundary conditions and peridynamics was analyzed. In order to eliminate this incompatibility, we proposed a new peridynamic motion equation in which the effects of boundary traction and boundary displacement constraint were introduced. The new peridynamic motion equation is invariant under the transformations of rigid translation and rotation. Meanwhile, it also satisfies the requirements of total linear and angular momentum equilibrium. By this motion equation, three kinds of boundary value problems containing the displacement boundary condition, the traction boundary condition and mixed boundary condition are characterized in peridynamics. As examples, we calculated static tension and longitudinal vibration of a finite rod. The acquired solutions exhibit obvious nonlocal features, and the vibration has the dispersion similar to one dimensional atom chain vibration.

Center of mass of a system of variable mass particles

Abstract We study the dynamics of the center of mass (CM) of a set of particles when the number of particles in the system varies during a short interval of time, or when the value of the mass of one of the particles depends on time. We show that in both situations, while the population or mass of the system varies, the linear momentum of the CM differs from the total linear momentum of the system. During this time interval, the dynamics of the CM is driven by the external forces that act on the system plus transient forces that depend on the position and velocity of the CM. This dynamics is applied to study the time evolution of the vector position of the CM of a system of free particles moving on an air rail in which the mass of one the particles varies in a short time interval. We want to show it to students that with the knowledge and tools they have at an elementary Mechanics course they can formulate and answer questions which are not in the textbooks.

Conservative Taylor least squares reconstruction with application to material point methods

SummaryWithin the standard material point method (MPM), the spatial errors are partially caused by the direct mapping of material‐point data to the background grid. In order to reduce these errors, we introduced a novel technique that combines the least squares method with the Taylor basis functions, called the Taylor least squares (TLS), to reconstruct functions from scattered data while preserving their integrals. The TLS technique locally approximates quantities of interest such as stress and density, and when used with a suitable quadrature rule, it conserves the total mass and linear momentum after transferring the material‐point information to the grid. The integration of the technique into MPM, dual domain MPM, and B‐spline MPM significantly improves the results of these methods. For the considered examples, the TLS function reconstruction technique resembles the approximation properties of highly accurate spline reconstruction while preserving the physical properties of the standard algorithm.

Swimming in Curved Surfaces and Gauss Curvature

The Cartesian-Newtonian paradigm of mechanics establishes that, within an inertial frame, a body either remains at rest or moves uniformly on a line, unless a force acts externally upon it. This crucial assertion breaks down when the classical concepts of space, time and measurement reveal to be inadequate. If, for example, the space is non-flat, an effective translation might occur from rest in the absence of external applied force. In this paper we examine mathematically the motion of a small object or lizard on an arbitrary curved surface. Particularly, we allow the lizard’s shape to undergo a cyclic deformation due exclusively to internal forces, so that the total linear momentum is conserved. In addition to the fact that the deformation produces a swimming event, we prove –under fairly simplifying assumptions that such a translationis some what directly proportional to the Gauss curvature of the surface at the point where the lizardlies.

Efficient implementation of energy conservation for higher order finite elements with variational integrators

In this paper, we introduce a variational integrator for higher order finite element models. The solution quality of finite element simulation strongly depends on the approximation in time and space. A variational integrator is robust due to conservation of both the balance of total linear momentum and the balance of total angular momentum. It also binds the error in the balance of total energy. We extend the variational integrator to an integrator that conserves the balance of total energy as well. However, the approximation in time cannot mend the approximation error in space. Therefore, we use a higher order approximation in space to obtain a better solution compared to the real live model that should be simulated. We show different numerical examples with Dirichlet and Neumann boundaries. For the Dirichlet boundaries we use the Lagrange multiplier method and for the Neumann boundaries we introduce the forces on the nodes by constant pressure.