Newton's Dynamics Research Articles

Newton's algorithm for discrete classical dynamics.

A recent article in J. Chem. Phys. argues that the two algorithms, the velocity-Verlet and position-Verlet integrators, commonly used in Molecular Dynamics (MD) simulations, are different [L. Ni and Z. Hu, J. Chem. Phys. 161, 226101 (2024)]. However, not only are the two algorithms just different formulations of the same discrete algorithm, but also are other simple discrete algorithms used in MD simulations in the natural sciences. They are all reformulations of the discrete algorithm derived by Newton in 1687 in PropositionI in the very first part of his book Principia. The different reformulations of Newton's algorithm for discrete dynamics lead to identical discrete dynamics with the same invariances, momentum, angular momentum, and energy as Newton's analytical dynamics. Hundreds of thousands of MD simulations with Newton's discrete dynamics have appeared but unfortunately with many recorded errors for energies, potential energies, temperatures, and heat capacities. The public software for MD should be corrected.

Extraplanetary system under modified gravity

Abstract We investigated the properties of exoplanets using a modified gravity law. We then devised a gravitational formula that linked the radius, mass, semimajor axis, and stellar mass of the exoplanet. This formula is then validated by taking into account four different astronomical organizations (WASP, HAT, TOI, and HATS). These findings are compared to those in the exoplanet.edu database. The findings corroborated the observational data. The Einstein curvature theory was created by incorporating planet spin into Newton's planet dynamics. As a result, the spin of the planet is connected to its gravity.

Mass Models of the Milky Way and Estimation of Its Mass from the Gaia DR3 Data Set

We use data from the Gaia DR3 data set to estimate the mass of the Milky Way (MW) by analyzing the rotation curve in the range of distances 5 to 28 kpc. We consider three mass models: The first model adds a spherical dark matter (DM) halo, following the Navarro–Frenk–White (NFW) profile, to the known stellar components. The second model assumes that DM is confined to the Galactic disk, following the idea that the observed density of gas in the Galaxy is related to the presence of a more massive DM disk (DMD), similar to the observed correlation between DM and gas in other galaxies. The third model only uses the known stellar-mass components and is based on the Modified Newton Dynamics (MOND) theory. Our results indicate that the DMD model is comparable in accuracy to the NFW and MOND models and fits the data better at large radii where the rotation curve declines but has the largest errors. For the NFW model, we obtain a virial mass M vir = (6.5 ± 0.3) × 1011 M ⊙ with concentration parameter c = 14.5, which is lower than what is typically reported. In the DMD case, we find that the MW mass is M d = (1.6 ± 0.5) × 1011 M ⊙ with a disk’s characteristic radius of R d = 17 kpc.

The Development of a Four-Tier Diagnostic Test Based on Modern Test Theory in Physics Education

<p style="text-align: justify;">Diagnostic tests are generally two or three-tier and based on classical test theory. In this research, the Four-Tier Diagnostic Test (FTDT) was developed based on modern test theory to determine understanding of physics levels: scientific conception (SC), lack of knowledge (LK), misconception (MSC), false negatives (FN), and false positives (FP). The goals of the FTDT are to (a) find FTDT constructs, (b) test the quality of the FTDT, and (c) describe students' conceptual understanding of physics. The development process was conducted in the planning, testing, and measurement phases. The FTDT consists of four-layer multiple-choice with 100 items tested on 700 high school students in Yogyakarta. According to the partial credit models (PCM), the student's responses are in the form of eight categories of polytomous data. The results of the study show that (a) FTDT is built on the aspects of translation, interpretation, extrapolation, and explanation, with each aspect consisting of 25 items with five anchor items; (b) FTDT is valid with an Aiken's V value in the range of 0.85-0.94, and the items fit PCM with Infit Mean Square (INFIT MNSQ) of 0.77-1.30, item difficulty index of 0.12-0.38, and the reliability coefficient of Cronbach's alpha FTDT is 0.9; (c) the percentage of conceptual understanding of physics from large to small is LK type 2 (LK2), FP, LK type 1 (LK1), FN, LK type 3 (LK3), SC, LK type 4 (LK4), and MSC. The percentage sequence of MSC based on the successive material is momentum, Newton's law, particle dynamics, harmonic motion, work, and energy. In addition, failure to understand the concept sequentially is due to Newton's law, particle dynamics, work and energy, momentum, and harmonic motion.</p>

Exact and quasi-exact Newton dynamics of n − 1 and n bodies n − 1 of which have equal masses

For n,n≥3 gravitating bodies n−1 of which have equal masses planar exact and spatial quasi-exact solutions of the Newton equation of motion are found. Exact planar solutions are also found for this equation for n−1,n≥3 bodies with equal masses. These solutions describe a motion of the bodies with equal masses in such a way that their coordinates coincide with vertices of a regular polygon in horizontal planes while the remaining body is either immobile at the origin or moves along the vertical axis crossing the origin. The Weinstein and center Lyapunov theorems are utilized.

Rigidity of Newton dynamics

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set.In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”.Our main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.

Thermal expansion of FeNi Invar and zinc-blende CdTe from the view point of local structure

Thermal expansion of FeNi Invar and zinc-blende CdTe was investigated from the view point of local structure using the extended x-ray absorption fine structure (EXAFS) spectroscopic data and the path-integral effective classical potential (PIECP) Monte Carlo computational simulations. In this Review article, first the quantum statistical perturbation theory is intuitively described to see different character concerning thermal expansion between the quantum and classical theories. The diatomic Br2 molecule is employed as a simple example. Second, the PIECP theory is briefly described to note advantages and disadvantages of this simulation technique. Historical background is also discussed for the EXAFS investigation of thermal expansion based on the quantum statistical theories. The results of the FeNi Invar alloy are then summarized. The origin of zero thermal expansion in the FeNi alloy is ascribed to the so-called Invar effect that implies the variation of the electronic structure of Fe atoms depending on temperature. Zero thermal expansion at low temperature is however found to originate from the vibrational quantum effect. It is also noted that the interatomic distances of Fe-Fe, Fe-Ni, and Ni-Ni pairs are slightly but meaningfully different from each other, although the alloy exhibit a simple fcc crystal. Such a pair- dependent difference is also true for thermal expansion and we will discuss thermal expansion from the local point of view, which is interestingly different from the lattice thermal expansion significantly. Finally, the results of the zinc blende (or diamond) structure are presented. Although the origin of negative thermal expansion in these tetrahedral crystals is known as a result of classical vibrational anomaly within the Newton dynamics theory, the quantum statistical simulation is found to be essential to reproduce the negative thermal expansion of CdTe. It is emphasized that the vibrational quantum effect and classical anharmonicity are of great importance for the understanding of low-temperature thermal expansion as well as the elastic constants.

‘Like thermodynamics before Boltzmann.’ On the emergence of Einstein's distinction between constructive and principle theories

In a 1919 article for the Times of London, Einstein declared the relativity theory to be a ‘principle theory’, like thermodynamics, rather than a ‘constructive theory’, like the kinetic theory of gases. The present paper attempts to trace back the prehistory of this famous distinction. It provides a systematic overview of Einstein's repeated use of the relativity theory/thermodynamics analysis after 1905 and inserts it into its historical settings, the early reception of relativity against the background of fin de siècle electron theories. Einstein initially used the relativity theory/thermodynamics comparison to address a specific objection. In his 1905 relativity paper he had determined the velocity-dependence of the electron's mass by adapting Newton's particle dynamics to the relativity principle. However, according to many, this result was not admissible without making some assumption about the structure of the electron. Einstein replied that the relativity theory is similar to thermodynamics. Unlike the usual physical theories, it does not directly try to construct models of specific physical systems; it provides empirically motivated and mathematically formulated criteria for the acceptability of such theories. New theories can be obtained by modifying existing theories valid in limiting case so that they comply with such criteria. Einstein progressively transformed this line of the defense into a positive heuristics. Instead of directly searching for new theories, it is often more effective to search for conditions that constraint the number of possible theories. The constructive/principle theories opposition should be considered not only as abstract classification of theories, but also as Einstein's attempt to formulate a sort of ‘logic of discovery’. The paper argues that most of Einstein's scientific successes were obtained by following the principle strategy. Most of his failures happened when he was forced to fall back to the constructive strategy.

A scalable galerkin multigrid method for real-time simulation of deformable objects

We propose a simple yet efficient multigrid scheme to simulate high-resolution deformable objects in their full spaces at interactive frame rates. The point of departure of our method is the Galerkin projection which is simple to construct. However, a naïve Galerkin multigrid does not scale well for large and irregular grids because it trades-off matrix sparsity for smaller sized linear systems which eventually stops improving the performance. Given that observation, we design our special projection criterion which is based on skinning space coordinates with piecewise constant weights, to make our Galerkin multigrid method scale for high-resolution meshes without suffering from dense linear solves. The usage of skinning space coordinates enables us to reduce the resolution of grids more aggressively, and our piecewise constant weights further ensure us to always deal with reasonably-sparse linear solves. Our projection matrices also help us to manage multi-level linear systems efficiently. Therefore, our method can be applied to different optimization schemes such as Newton's method and Projective Dynamics, pushing the resolution of a real-time simulation to orders of magnitudes higher. Our final GPU implementation outperforms the other state-of-the-art GPU deformable body simulators, enabling us to simulate large deformable objects with hundred thousands of degrees of freedom in real-time.

A Model Problem for Optimal Control of a Parabolic PDE Fully Coupled to ODEs

AbstractIn many applications we encounter fully coupled systems involving a partial differential equation (PDE) and an ordinary differential equation (ODE). A particular class is the case in which the PDE is parabolic. This coupled optimal control problem is considered in theory in [1]. Here we consider a toy example with the heat equation and a second order ODE given by Newton dynamics. The acceleration in the ODE may be controlled and is subject to box constraints. We present numerical optimal control results for this toy example.

Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments

The possibility that ultra-diffuse galaxies are lacking dark matter has recently stimulated interest to check the validity of modified Newton dynamics (MOND) predictions on the scale of such galaxies. It has been shown that the external field effect (EFE) induced by the close-by galaxy can suppress the velocity dispersion of these systems, so that they appear almost dark matter free in the Newtonian context. Here, following up on this, we are making a priori predictions for the velocity dispersion of 22 ultra-diffuse galaxies in the nearby Universe. This sample can be used to test MOND and the EFE with future follow-up measurements. We have constructed a catalog of nearby ultra-diffuse galaxies in galaxy group environments, and set upper and lower limits for the possible velocity dispersion allowed in MOND, taking into account possible variations in the mass-to-light ratio of the dwarf and in the distance to the galaxy group. The prediction for the velocity dispersion is made as a function of the three dimensional separation of the dwarf to its host. In 17 out of 22 cases, the EFE plays a crucial role in the prediction.

Parallel Multigrid for Nonlinear Cloth Simulation

AbstractAccurate high‐resolution simulation of cloth is a highly desired computational tool in graphics applications. As single‐resolution simulation starts to reach the limit of computational power, we believe the future of cloth simulation is in multi‐resolution simulation. In this paper, we explore nonlinearity, adaptive smoothing, and parallelization under a full multigrid (FMG) framework. The foundation of this research is a novel nonlinear FMG method for unstructured meshes. To introduce nonlinearity into FMG, we propose to formulate the smoothing process at each resolution level as the computation of a search direction for the original high‐resolution nonlinear optimization problem. We prove that our nonlinear FMG is guaranteed to converge under various conditions and we investigate the improvements to its performance. We present an adaptive smoother which is used to reduce the computational cost in the regions with low residuals already. Compared to normal iterative solvers, our nonlinear FMG method provides faster convergence and better performance for both Newton's method and Projective Dynamics. Our experiment shows our method is efficient, accurate, stable against large time steps, and friendly with GPU parallelization. The performance of the method has a good scalability to the mesh resolution, and the method has good potential to be combined with multi‐resolution collision handling for real‐time simulation in the future.

Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, , we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N − 1 integrals of motion, notably, their scaling with N, and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs–Boltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.

Multiscale Multilevel Approach to Solution of Nanotechnology Problems

The paper is devoted to a multiscale multilevel approach for the solution of nanotechnology problems on supercomputer systems. The approach uses the combination of continuum mechanics models and the Newton dynamics for individual particles. This combination includes three scale levels: macroscopic, mesoscopic and microscopic. For gas–metal technical systems the following models are used. The quasihydrodynamic system of equations is used as a mathematical model at the macrolevel for gas and solid states. The system of Newton equations is used as a mathematical model at the mesoand microlevels; it is written for nanoparticles of the medium and larger particles moving in the medium. The numerical implementation of the approach is based on the method of splitting into physical processes. The quasihydrodynamic equations are solved by the finite volume method on grids of different types. The Newton equations of motion are solved by Verlet integration in each cell of the grid independently or in groups of connected cells. In the framework of the general methodology, four classes of algorithms and methods of their parallelization are provided. The parallelization uses the principles of geometric parallelism and the efficient partitioning of the computational domain. A special dynamic algorithm is used for load balancing the solvers. The testing of the developed approach was made by the example of the nitrogen outflow from a balloon with high pressure to a vacuum chamber through a micronozzle and a microchannel. The obtained results confirm the high efficiency of the developed methodology.

1.ケプラー型ニュートン力学 : マイケルソンは光の絶対速度を検出していた(大会テーマ「新たな理科教育の創造」)

1.ケプラー型ニュートン力学 : マイケルソンは光の絶対速度を検出していた(大会テーマ「新たな理科教育の創造」)

Relativistic Newtonian Dynamics under a central force

Planck's formula and General Relativity indicate that potential energy influences spacetime. Using Einstein's Equivalence Principle and an extension of his Clock Hypothesis, an explicit description of this influence is derived. We present a new relativity model by incorporating the influence of the potential energy on spacetime in Newton's dynamics for motion under a central force. This model extends the model used by Friedman and Steiner (EPL, 113 (2016) 39001) to obtain the exact precession of Mercury without curving spacetime. We also present a solution of this model for a hydrogen-like atom, which explains the reason for a probabilistic description.

Two-scale computation of N2–H2 jet flow based on QGD and MMD on heterogeneous multi-core hardware

Gas mixture flows take place in a wide range of scientific and technical problems, including space applications, microchannel streams, nanotechnology and others. This paper deals with numerical simulation of gas mixture flows in microchannels. For such flows speeds can have supersonic, subsonic and transonic ranges according to parameters of gas stream and properties. The supersonic expansion of gas mixtures is accompanied by several simultaneous nonequilibrium processes: interfusion of layers, shock waves, heat exchange with walls etc. To investigate these complex problems, it is essential to develop adequate numerical technique that contains stable and robust numerical algorithms and calculations based on fine grids. These calculations are supposed to be carried out by means of high performance computer systems.This paper presents a new method for simulating gas mixture flows that enables us to solve similar issues and reduce computing time (Garzo et al., 1989; Elizarova, 2009). Particularity of our approach is combination of continuum mechanics and Newton's dynamics models. For computer realization of the approach we use grid methods for solution of macroscopic quasigasdynamic (QGD) equations and molecular dynamics methods (MMD). Some results of numerical modelling are discussed.

A HANDY SPECTROGRAPH AND ITS APPLICATION IN ASTRONOMICAL EDUCATION

I use a common educational spectrographic device (SV2100R) in order to obtain astronomical spec-tra after inventing a new adaptor for telescopes. Experimental classes and learning projects in schoolsand public outreach are well established regarding imaging and photometry observations. However, ex-periments using astronomical spectrographs are rather hard to nd because the procedures of spectralextraction and wavelength calibration is less convenient. SV2100R is a 1D CCD array and thus has theadvantage of not requiring spectral extraction. In addition, basic wavelength calibration is preformed bythe the provided software. It was adapted to a 12-inch reecting telescope in the Korea Science Academyof KAIST in Busan and a spectrum of the bright object, Arcturus, was successfully obtained. This meansone can provide educational programs on the topic of astronomical spectra. A few suggested projects arepresented.Key words: Spectroscopy; adaptor; education1. INTRODUCTIONLight gives important information and spectra play akey role in investigating stellar temperatures, chemicalcompositions and relative motions. Electromagnetic ra-diation is one of main objectives of education in as-tronomy as well as for Newton dynamics. Introduc-tory astronomy often teaches about spectral properties,Doppler shift, Wien’s law, Stefan-Boltzman’s law, andKirchho ’s law (Bardar & Brecher 2008; Slater et al.2001).Major learning objectives are 1) having an intrinsicinterest in science, 2) developing critical reasoning skillsand 3) understanding nature. Self-motivated learn-ing is thought to be an e ective learning process andproject based education is becoming important, as wellas lecture-based methods. Handling telescopes in astro-nomical observatories is a good tool for active learning inastronomy. Most educational programs using telescopesfor imaging or photometric observations at universities,schools for science and public outreach in astronomi-cal centers. Understanding and experiments for electro-magnetic spectra have an important role in educationfor introductory astronomy. The development of spec-troscopic experiments using telescope will be helpful fordeveloping knowledge of light and electromagnetic spec-tra for both introductory astronomy and self-motivatedscience projects.In schools, universities and public outreach programs,the prism or grating lm is normally applied to ob-http://pkas.kas.orgserve the spectrum of light sources. It is straightfor-ward, however it is not recordable. If the grating lm iscombined with a DSLR camera, the spectrum could beeasily recorded and spectra extracted. In this case, spec-tral extraction and wavelength calibration are requiredfor data reduction. Recently, an astronomical spectro-scopic device has been made for education in terms oflearning the structure of the device and the detection ofastronomical spectra Lee et al. (2013).As a result of the importance of spectral educationin astronomy, I developed astronomical spectroscopicobservations in education using a commercial spectro-graph attached to the telescope using a new adaptor.The spectroscope has been widely used for educationalpurposes, and is normally used in chemistry projects orlabs. The new adaptor xes the spectroscope to the tele-scope and makes detection through the telescope easy.It has obtained spectra of reected sunlight from brightplanets and bright stellar spectra.2. SPECTROSCOPIC DEVICE WITH NEW ADAPTORThe educational spectroscopic device, SV2100R, manu-factured by K-Mac, has been used. It contains a grat-ing of 600 lines/mm and uses a 1-D CCD array of 2,048pixels. Wavelength coverage is 250 ˘ 800 nm and hasa spectral resolution better than than 1nm. There is noneed to extract spectra from the spectral image becauseit is already a 1D array. The provided software makesit easier for to carry out reduction in terms of spectralextraction and wavelength calibration. Due to the 1Darray, there is no need to extract the spectrum from the757

Coupling the nongravitational forces and modified Newton dynamics for cometary orbits

In recent work (Milgrom 2009, Blanchet & Novak 2011), the authors showed that MOdified Newton Dynamics (MOND) have a non-negligible secular perturbation effect on planets with large semi-major axes (gaseous planets) in the Solar System. Some comets also have a very eccentric orbit with a large semi-major axis (Halley family comets) going far away from the Sun (more than 15 AU) in a low acceleration regime where they would be subject to MOND perturbation. They also approach the Sun very closely (less than 3 AU) and are affected by the sublimation of ices from their nucleus, triggering so-called non-gravitational forces. The main goal of this paper is to investigate the effect of MOND perturbation on three comets with various orbital elements (2P/Encke, 1P/Halley and 153P/Ikeya-Zhang) and then compare it to the non-gravitational perturbations. It is motivated by the fact that when fitting an outgassing model for a comet, we have to take into account all of the small perturbing effects to avoid absorbing these effects into the non-gravitational parameters. Otherwise, we could derive a completely wrong estimation of the outgassing. For this work, we use six different forms of MOND functions and compute the secular variations of the orbital elements due to MOND and non-gravitational perturbations. We show that, for comets with large semi-major axis, the MONDian effects are not negligible compared to the non-gravitational perturbations.

A Change of the Gravitational Interaction on the Galactic Distances

In order to explain anomalous velocities of star rotation in galaxies, some researchers propose corrections to Newton's dynamics. In contrast to this opinion, we suppose that it is possible to correct the potential of gravity interaction. The modulus of the force of interaction between the bodies may be written as: F=Exp (-R2/α)γMm/R2 + (1 - Exp(-R2/α))δMm/R, где α ~ 4×105 (au)-2, δ ~ 2.7×10-31 N m kg-2. This correction allows one to conserve the force of attraction between the bodies within the Solar System proportional to 1/R2, while for longer interstellar distances it will be proportional to 1/R. It provides a good description both for the motion of planets in the Solar System and for the motions of stars and galaxies. Within the framework of the proposed approach, there is no necessity to introduce the idea of dark matter. According to the analysis of the proposed interaction with the help of the virial theorem, it does not contradict basic notions on the motion of stars and planets; in particular, it does not contradict the established accelerated expansion of the Universe.