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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Phylogenomics and diversification of Sordariomycetes

The Sordariomycetes is a specious, morphologically diverse, and widely distributed class of the phylum Ascomycota that forms a well-supported clade diverged from Leotiomycetes. Aside from their ecological significance as plant and human pathogens, saprobes, endophytes, and fungicolous taxa, species of Sordariomycetes produces a wide range of chemically novel and diverse metabolites used in important fields. Recent phylogenetic analyses derived from a small number of genes have considerably increased our understanding of the family, order, and subclass relationships within Sordariomycetes, but several important groups have not been resolved well. In addition, there are various paraphyletic or polyphyletic groups. Moreover, the criteria used to establish higher ranks remain highly variable across different studies. Therefore, the taxonomy of Sordariomycetes is in constant flux, remains poorly understood, and is subject to much controversy. Here, for the first time, we have assembled a phylogenetic dataset containing 638 genomes representing the 156 genera, 50 families, and 17 orders and 5 subclasses of Sordariomycetes. This data set is based on 1124 genes and results in a well-resolved phylogenomic tree. We further constructed an evolutionary timeline of Sordariomycetes diversification based on the genomic data sets. Our divergence time estimate results are inconsistent with previous studies, suggesting estimates of node ages are less precise and varied. Based on these results, we discuss the higher ranks of Sordariomycetes and empirically propose an unprecedented taxonomic framework for the class.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2015

Integrated Assessments of the Impact of Climate Change on Agriculture: An Overview of AgMIP Regional Research in South Asia

South Asia encompasses a wide and highly varied geographic region, and includes climate zones ranging from the mountainous Himalayan territory to the tropical lowland and coastal zones along alluvial floodplains. The region's climate is dominated by a monsoonal circulation that heralds the arrival of seasonal rainfall, upon which much of the regional agriculture relies. The spatial and temporal distribution of this rainfall is, however, not uniform over the region. Northern South Asia, central India, and the west coast receive much of their rainfall during the southwest monsoon season, between June and September. These rains partly result from the moisture transport accompanying the monsoonal winds, which move in the southwesterly direction from the equatorial Indian Ocean. Regions further south, such as south/southeast India and Sri Lanka, may receive rains from both the southwest monsoon, and also during the northeast monsoon season between October and December (with northeasterly monsoon wind flow and moisture flux), which results in a bi- or multi-modal rainfall distribution. In addition, rainfall across South Asia displays a large amount of intraseasonal and interannual variability. Interannual variability is influenced by many drivers, both natural (e.g., El Ni-Southern Oscillation; ENSO) and man-made (e.g., rising temperatures due to increasing greenhouse gas concentrations), and it is challenging to obtaining accurate time-series of annual rainfall, even amongst various observed data products, which display inconsistencies amongst themselves. These climatic and rainfall variations can further complicate South Asia's agricultural and water management. Agriculture employs at least 65 of the workforce in most South Asian countries, and nearly 80 of South Asia's poor inhabit rural areas. Understanding the response of current agricultural production to climate variability and future climate change is of utmost importance in securing food and livelihoods for South Asia's growing population. In order to assess the future of food and livelihood security across South Asia, the Agricultural Model Intercomparison and Improvement Project (AgMIP) has undertaken integrated climate-crop-economic assessments of the impact of climate change on food security and poverty in South Asia, encompassing Bangladesh, India, Nepal, Pakistan, and Sri Lanka. AgMIP has funded, on a competitive basis, four South Asian regional research teams (RRTs) and one South Asian coordination team (CT) to undertake climate-crop-economic integrated assessments of food security for many districts in each of these countries, with the goal of characterizing the state of food security and poverty across the region, and projecting how these are subject to change under future climate change conditions.

mountains↗
DOE OSTI2021

A Nonstationary and Non-Gaussian Moving Average Model for Solar Irradiance

Historically, power has flowed from large power plants to customers. Increasing penetration of distributed energy resources such as solar power from rooftop photovoltaic has made the distribution network a two-way-street with power being generated at the customer level. The incorporation of renewables introduces additional uncertainty and variability into the power grid. Distribution network operation studies are being adapted to include renewables; however, such studies require high quality solar irradiance data that adequately reflect realistic meteorological variability. Data from satellite-based products are spatially complete, but temporally coarse, whereas solar irradiances exhibit high frequency variation at very fine timescales. We propose a new stochastic method for temporally downscaling global horizontal irradiance (GHI) to 1 min resolution, but we do not consider the spatial aspect due to limited availability of the in situ irradiance measurements. Solar irradiance's first and second-order structures vary diurnally and seasonally, and our model adapts to such nonstationarity. Empirical irradiance data exhibits highly non-Gaussian behavior; we develop a nonstationary and non-Gaussian moving average model that is shown to capture realistic solar variability at multiple timescales. We also propose a new estimation scheme based on Cholesky factors of empirical autocovariance matrices, bypassing difficult and inaccessible likelihood-based approaches. The model is demonstrated for a case study of three locations that are located in diverse climates through the United States. The model is compared against competitors from the literature and is shown to provide better uncertainty and variability quantification on testing data.

Cholesky factor↗
NASA NTRS2005

Global Biomass Variation and its Geodynamic Effects, 1982-1998

Redistribution of mass near Earth's surface alters its rotation, gravity field, and geocenter location. Advanced techniques for measuring these geodetic variations now exist, but the ability to attribute the observed modes to individual Earth system processes has been hampered by a shortage of reliable global data on such processes, especially hydrospheric processes. To address one aspect of this deficiency, 17 yrs of monthly, global maps of vegetation biomass were produced by applying field-based relationships to satellite-derived vegetation type and leaf area index. The seasonal variability of biomass was estimated to be as large as 5 kg m(exp -2). Of this amount, approximately 4 kg m(exp -2) is due to vegetation water storage variations. The time series of maps was used to compute geodetic anomalies, which were then compared with existing geodetic observations as well as the estimated measurement sensitivity of the Gravity Recovery and Climate Experiment (GRACE). For gravity, the seasonal amplitude of biomass variations may be just within GRACE'S limits of detectability, but it is still an order of magnitude smaller than current observation uncertainty using the satellite-laser-ranging technique. The contribution of total biomass variations to seasonal polar motion amplitude is detectable in today's measurement, but it is obscured by contributions from various other sources, some of which are two orders of magnitude larger. The influence on the length of day is below current limits of detectability. Although the nonseasonal geodynamic signals show clear interannual variability, they are too small to be detected.

Rodell, M.↗
DOE OSTI2023

A multifidelity deep operator network approach to closure for multiscale systems

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and/or accuracy of the multifidelity-based closure, we employ the recently developed ``in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging for the two-dimensional Navier-Stokes equations. Furthermore, the numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

42 ENGINEERING↗
NASA NTRS2001

Discontinuous Galerkin Methods for NonLinear Differential Systems

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Barth, Timothy↗
NASA NTRS1990

The ultraviolet and visible light variability of BP Tauri - Possible clues for the origin of T Tauri star activity

BP Tau is a moderate-strength T Tauri star, which has previously been studied photometrically by Vrba et al. (1986, 1989) and shown to have a 7.6-day light cycle. Here, BP Tau was reobserved in order to monitor this star with the IUE satellite. In conjunction with the UV spectroscopy, nearly simultaneous ground-based optical photometry was also obtained. At optical wavelengths, the star was initially found to be varying with a period of 7.6 days. This variability suddenly vanished and was followed by a quiescent state that lasted about 11 days. The star then resumed its variability with a period of about 6.1 days. The UV continuum and most of the UV emission lines appeared to vary in phase with the optical flux. No variations were observed on the very short timescales ranging from several hours to half a day, expected on the basis of axisymmetric, quasi-steady-state accretion disk models of T Tauri stars. However, the nearly week-long variations that were observed may be consistent with a magnetically buffered accretion model, which has been proposed by Uchida and Shibata.

Simon, Theodore↗
NASA NTRS1995

Characterization of High Ge Content SiGe Heterostructures and Graded Alloy Layers Using Spectroscopic Ellipsometry

Si(x)Ge(1-x)heterostructures on Si substrates have been widely studied due to the maturity of Si technology. However, work on Si(x)Ge)1-x) heterostructures on Ge substrates has not received much attention. A Si(x)Ge(1-x) layer on a Si substrate is under compressive strain while Si(x)Ge(1-x) on Ge is under tensile strain; thus the critical points will behave differently. In order to accurately characterize high Ge content Si(x)Ge(1-x) layers the energy shift algorithm used to calculate alloy compositions, has been modified. These results have been used along with variable angle spectroscopic ellipsometry (VASE) measurements to characterize Si(x)Ge(1-x)/Ge superlattices grown on Ge substrates. The results agree closely with high resolution x-ray diffraction measurements made on the same samples. The modified energy shift algorithm also allows the VASE analysis to be upgraded in order to characterize linearly graded layers. In this work VASE has been used to characterize graded Si(x)Ge(1-x) layers in terms of the total thickness, and the start and end alloy composition. Results are presented for a 1 micrometer Si(x)Ge(1-x) layer linearly graded in the range 0.5 less than or equal to x less than or equal to 1.0.

Heyd, A. R.↗
NASA NTRSDate not supplied

Powder Bed Fusion Laser Beam Metals Additive Manufacturing: Process Monitoring Approaches for Qualification and Certification

The use of in-situ process monitoring is of interest to lower the cost of inspection for the qualification of powder bed fusion laser beam metal (PBF-LB/M) additively manufactured (AM) parts. Precise monitoring of the PBF-LB/M AM build process constitutes a multi-scale and multi-discipline task. There are several significant challenges to the in-situ approach: the synchronization of sensor signals to process steps; the physical interpretation and classification of sensor signals; managing very large datasets; and comparing the inputs with the observed monitoring signals. At NASA Langley Research Center, a configurable architecture additive testbed has been developed to monitor the build process with synchronized sensors. The philosophy and method adopted for the synchronization of the cameras with laser power and position throughout a complex PBF-LB/M AM build will be described. The synchronized in-situ monitoring signals are compared with ex-situ nondestructive inspection, x-ray computed tomography (XCT). Such comparisons permit a better understanding of how the sequential process actions of LPBF-AM can affect build quality. The multi-scale and complex process of printing additively manufactured (AM) parts can have unexpected, but predictable, build conditions that result in material microstructure variability. This presentation will describe an additive manufacturing model-based process metric (AM-PM) computational method that is a fully parallel reduced order modeling approach developed to evaluate the evolution of AM processes. This method couples the known sequence of the AM process with a physically informed nearest neighbors’ calculation to map the conditions of a part-scale build. The result is a map of the build that is derived directly from build files or in-situ process monitoring sensors. The methodology of the approach will be described and mapped to the porosity observed from XCT for a complex PBF-LB/M build. Such comparative results develop understanding of how the sequential process actions can affect the PBF-LB/M AM build quality and microstructure variability.

Laser Powder Bed Fusion↗
DOE OSTI2022

Predicting Rare Earth Element Potential in Produced and Geothermal Waters of the United States via Emergent Self-Organizing Maps

This work applies emergent self-organizing map (ESOM) techniques, a form of machine learning, in the multidimensional interpretation and prediction of rare earth element (REE) abundance in produced and geothermal waters in the United States. Visualization of the variables in the ESOM trained using the input data shows that each REE, with the exception of Eu, follows the same distribution patterns and that no single parameter appears to control their distribution. Cross-validation, using a random subsample of the starting data and only using major ions, shows that predictions are generally accurate to within an order of magnitude. Using the same approach, an abridged version of the U.S. Geological Survey Produced Waters Database, Version 2.3 (which includes both data from produced and geothermal waters) was mapped to the ESOM and predicted values were generated for samples that contained enough variables to be effectively mapped. Results show that in general, produced and geothermal waters are predicted to be enriched in REEs by an order of magnitude or more relative to seawater, with maximum predicted enrichments in excess of 1000-fold. Cartographic mapping of the resulting predictions indicates that maximum REE concentrations exceed values in seawater across the majority of geologic basins investigated and that REEs are typically spatially co-associated. The factors causing this co-association were not determined from ESOM analysis, but based on the information currently available, REE content in produced and geothermal waters is not directly controlled by lithology, reservoir temperature, or salinity.

Engle, Mark A. (ORCID:0000000152587374)↗
DOE OSTI2020

Electric Power Grid and Natural Gas Network Operations and Coordination

In this report, we present findings from three studies related to the coordination of natural gas and electricity system operations. We first propose and demonstrate a modeling platform for examining the interdependence of natural gas and electricity networks based on a direct current unit-commitment and economic dispatch model for the power system and a transient hydraulic gas model for the gas system. We use this platform to analyze the value of day-ahead coordination of power and natural gas network operations and to show the importance of considering gas system constraints when analyzing power systems operation with high penetration of gas generators and variable renewable energy sources. In the second study, we utilize our modeling platform to consider the U.S. Federal Energy Regulatory Commission (FERC) Order 809, issued in 2015 to improve day-ahead and intraday coordination of power and gas systems. Finally, in the third study we expand our modeling platform to focus on market-based coordination of electricity and natural gas system operations for a real system, namely a subset of the power and gas networks in the Front Range region of Colorado. We use real system data to evaluate the benefits of coordination operations under different conditions, including different levels of renewable penetration and the use of time-variant, shaped flow nominations. Our results indicate that coordination at various timescales can contribute to a reduction in curtailed gas in high-stress periods (such as those with large ramps in gas offtakes) and a reduction in energy consumption of gas compressor stations. We find that intraday coordination can reduce total power system production costs and natural gas deliverability constraints, yielding cost and reliability benefits. We observe these benefits for the test system as well as in the Colorado case study, where we find that coordination and shaped flows may provide additional value for systems with high penetration of variable renewable energy. Together, these three studies demonstrate a pathway for integrated gas and electricity grid modeling and for studying the benefits of coordinated operations of these increasingly interdependent energy systems.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2017

Application of high-order numerical schemes and Newton-Krylov method to two-phase drift-flux model

This study concerns the application and solver robustness of the Newton-Krylov method in solving two-phase flow drift-flux model problems using high-order numerical schemes. In our previous studies, the Newton-Krylov method has been proven as a promising solver for two-phase flow drift-flux model problems. However, these studies were limited to use first-order numerical schemes only. Moreover, the previous approach to treating the drift-flux closure correlations was later revealed to cause deteriorated solver convergence performance, when the mesh was highly refined, and also when higher-order numerical schemes were employed. In this study, a second-order spatial discretization scheme that has been tested with two-fluid two-phase flow model was extended to solve drift-flux model problems. In order to improve solver robustness, and therefore efficiency, a new approach was proposed to treating the mean drift velocity of the gas phase as a primary nonlinear variable to the equation system. With this new approach, significant improvement in solver robustness was achieved. With highly refined mesh, the proposed treatment along with the Newton-Krylov solver were extensively tested with two-phase flow problems that cover a wide range of thermal-hydraulics conditions. Satisfactory convergence performances were observed for all test cases. Numerical verification was then performed in the form of mesh convergence studies, from which expected orders of accuracy were obtained for both the first-order and the second-order spatial discretization schemes. Finally, the drift-flux model, along with numerical methods presented, were validated with three sets of flow boiling experiments that cover different flow channel geometries (round tube, rectangular tube, and rod bundle), and a wide range of test conditions (pressure, mass flux, wall heat flux, inlet subcooling and outlet void fraction).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2016

Streamflow variability over the 1881–2011 period in northern Quebec: comparison of hydrological reconstructions based on tree rings and geopotential height field reanalysis

Over the last decades, different methods have been used by hydrologists to extend observed hydro-climatic time series, based on other data sources, such as tree rings or sedimentological datasets. For example, tree ring multi-proxies have been studied for the Caniapiscau Reservoir in northern Québec (Canada), leading to the reconstruction of flow time series for the last 150 years. In this paper, we applied a new hydro-climatic reconstruction method on the Caniapiscau Reservoir and compare the obtained streamflow time series against time series derived from dendrohydrology by other authors on the same catchment and study the natural streamflow variability over the 1881–2011 period in that region. This new reconstruction is based not on natural proxies but on a historical reanalysis of global geopotential height fields, and aims firstly to produce daily climatic time series, which are then used as inputs to a rainfall–runoff model in order to obtain daily streamflow time series. The performances of the hydro-climatic reconstruction were quantified over the observed period, and showed good performances, in terms of both monthly regimes and interannual variability. The streamflow reconstructions were then compared to two different reconstructions performed on the same catchment by using tree ring data series, one being focused on mean annual flows and the other on spring floods. In terms of mean annual flows, the interannual variability in the reconstructed flows was similar (except for the 1930–1940 decade), with noteworthy changes seen in wetter and drier years. For spring floods, the reconstructed interannual variabilities were quite similar for the 1955–2011 period, but strongly different between 1880 and 1940. Here, the results emphasize the need to apply different reconstruction methods on the same catchments. Indeed, comparisons such as those above highlight potential differences between available reconstructions and, finally, allow a retrospective analysis of the proposed reconstructions of past hydro-climatological variabilities.

58 GEOSCIENCES↗
NASA NTRS2013

A Stochastic Fractional Dynamics Model of Space-time Variability of Rain

Rainfall varies in space and time in a highly irregular manner and is described naturally in terms of a stochastic process. A characteristic feature of rainfall statistics is that they depend strongly on the space-time scales over which rain data are averaged. A spectral model of precipitation has been developed based on a stochastic differential equation of fractional order for the point rain rate, that allows a concise description of the second moment statistics of rain at any prescribed space-time averaging scale. The model is thus capable of providing a unified description of the statistics of both radar and rain gauge data. The underlying dynamical equation can be expressed in terms of space-time derivatives of fractional orders that are adjusted together with other model parameters to fit the data. The form of the resulting spectrum gives the model adequate flexibility to capture the subtle interplay between the spatial and temporal scales of variability of rain but strongly constrains the predicted statistical behavior as a function of the averaging length and times scales. We test the model with radar and gauge data collected contemporaneously at the NASA TRMM ground validation sites located near Melbourne, Florida and in Kwajalein Atoll, Marshall Islands in the tropical Pacific. We estimate the parameters by tuning them to the second moment statistics of radar data. The model predictions are then found to fit the second moment statistics of the gauge data reasonably well without any further adjustment.

TRMM↗
NASA NTRS1998

Deep and Monitoring Observations of the Black Hole Candidates 1E 1740.7-2942 and GRS 1758-258

The purpose of this effort was to obtain detailed information on the long term variability of the power spectra and energy spectra of the two Black Hole Candidates (BHCs) and so-called "micro-quasars", 1E 1740.7-2942 and GRS 1758-258. Observations with the pointed instruments on the Rossi X-ray Timing Explorer (RXTE) were carried out on a weekly basis for an entire year, in order to observe the extent to which these sources changed on long timescales. The observations also served as a trigger for longer observations carried out under a sister proposal "Target of Opportunity Observations of the Black Hole Candidates 1E 1740.7-2942 and GRS 1758-258". The work done at UCSD by W. Heindl consisted first of monitoring the data from the High Energy X-ray Timing Experiment (HEXTE) to determine when the sources were bright enough to trigger our Target of Opportunity observations. He was then responsible for the reduction and interpretation of the HEXTE data and for contributing to the publication of results of this work. Historically, these objects have been highly variable on long timescales. Observations between 1988 and 1995 found that they tend to spend periods of several months in high and low flux states which differ in luminosity by more than an order of magnitude. In more than 2 years of RXTE observations to date, we have found variability only on the level of tens of percent. Both sources have remained near their historical maximum luminosities during this time. This is a significant change from previous behavior, and indicates that their accretion rates have stabilized in recent years. In addition, their observed spectra have been quite stable.

Heindl, William A.↗
DOE OSTI2020

Localized basal area affects soil respiration temperature sensitivity in a coastal deciduous forest

Abstract. Soil respiration (Rs), the flow of CO2 from the soil surface to the atmosphere, is one of the largest carbon fluxes in the terrestrial biosphere. The spatial variability of Rs is both large and poorly understood, limiting our ability to robustly scale it in space. One factor in Rs spatial variability is the autotrophic contribution from plant roots, but it is uncertain how the presence of plants affects the magnitude and temperature sensitivity of Rs. This study used 1 year of Rs measurements to examine the effect of localized basal area on Rs in the growing and dormant seasons, as well as during moisture-limited times, in a temperate, coastal, deciduous forest in eastern Maryland, USA. In a linear mixed-effects model, tree basal area within a 5 m radius (BA5) exerted a significant positive effect on the temperature sensitivity of soil respiration. Soil moisture was the dominant control on Rs during the dry portions of the year, while soil moisture, temperature, and BA5 all exerted significant effects on Rs in wetter periods. Our results suggest that autotrophic respiration is more sensitive to temperature than heterotrophic respiration at these sites, although we did not measure these source fluxes directly, and that soil respiration is highly moisture sensitive, even in a record-rainfall year. The Rs flux magnitudes (0.46–15.0 µmol m-2 s-1) and variability (coefficient of variability 10 %–23 % across plots) observed in this study were comparable to values observed in similar forests. Six Rs observations would be required in order to estimate the mean across all study sites to within 50 %, and 518 would be required in order to estimate it to within 5 %, with 95 % confidence. A better understanding of the spatial interactions between plants and microbes, as well as the strength and speed of above- and belowground coupling, is necessary to link these processes with large-scale soil-to-atmosphere C fluxes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Three-point energy correlators in hadronic Higgs boson decays

We present the analytic calculation of the leading-order three-point energy correlator (EEEC) in hadronic Higgs decays, including both the gluon-initiated channel H → g g + X and quark-initiated channel H → q q ¯ + X . The phase-space integration is evaluated directly using Mandelstam variables s i j = ( p i + p j ) 2 , and the appearing square roots can be rationalized by either conformal ratios or celestial coordinate variables. Throughout the calculation, we observe the same transcendental function space as in N = 4 super–Yang-Mills theory and e + e − → hadrons . Different infrared limits are also explored using the full analytic result, offering the fixed-order data for EEEC factorization and resummation. Given its nontrivial shape dependence, the EEEC presents an excellent opportunity to explore the dynamics of gluon jets originating from the H → g g decay channel at future lepton colliders. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2016

Off-shell hydrodynamics from holography

In this article, we outline a program for obtaining an action principle for dissipative fluid dynamics by considering the holographic Wilsonian renormalization group applied to systems with a gravity dual. As a first step, in this paper we restrict to systems with a non-dissipative horizon. By integrating out gapped degrees of freedom in the bulk gravitational system between an asymptotic boundary and a horizon, we are led to a formulation of hydrodynamics where the dynamical variables are not standard velocity and temperature fields, but the relative embedding of the boundary and horizon hypersurfaces. At zeroth order, this action reduces to that proposed by Dubovsky et al. as an off-shell formulation of ideal fluid dynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗