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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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DOE OSTI2021

Assessment of Updraft Modeling Bias Using Computational Fluid Dynamics

Golden Eagle (Aquila chrysaetos) habitats may overlap with wind energy development in some regions of the US. Eagles, and similar soaring bird species, are therefore at risk of collision with wind turbines when flying through wind farms. Recently developed behavioral modeling approaches can predict the presence of eagles near turbines within the rotor-swept layer but require reliable prediction of atmospheric flowfield conditions. In particular, the vertical component of the wind speed dictates a soaring bird's ability to maintain or gain altitude, since they rely on updrafts to subsidize their flight. In this work, we investigate the atmospheric conditions around a wind farm in complex terrain and compare methods for atmospheric characterization. We use computational fluid dynamics (specifically, large-eddy simulations, or LES) to simulate the atmospheric boundary layer over a region encompassing multiple wind farms with high temporal and spatial resolution (seconds and 10's of meters, respectively). We compare traditional non-simulation-based methods of determining the orographic updraft potential based on wind direction, terrain slope and aspect, with the flowfields from LES that include both orographic updrafts alone and combined thermal and orographic updrafts. Preliminary analysis suggests that although the model captures the horizontal pattern of vertical updrafts, their magnitude can be improved with information about the surface heat flux, which is usually correlated with time of the day. Within our study region, we found that the low-fidelity model may over- or underestimate updraft potential by up to 400% at 80 m AGL, depending on local orographic features. This can result in an inaccurate representation of eagle presence and, consequently, risk. Another important finding is that flowfield time-averaging can hide important details about the flight environment, including how thermally generated flow structures within the atmospheric boundary layer (e.g., convective rolls and/or cells) may be important drivers of eagle flight.

atmospheric turbulence↗

DOE OSTI2020

A multi-dimensional dataset for two-phase instability in low pressure natural circulation based on direct transient local measurement

Thermal hydraulic instability can challenge the control and the safety of nuclear reactors. This experimental research captures and investigates the instability in a low-pressure (1–10 atmospheres) natural circulation loop with an adiabatic chimney installed above the heated section. Direct transient local measurement of two-phase flow is performed at multiple axial locations along the flow channel with thermocouples and conductivity probes traversed by customized high-precision mechanisms. Based on the periodicity of oscillations, a post-processing method is proposed to align the data captured non-simultaneously to reconstruct the instability behavior in one representative period. The averaging operation over multiple oscillation cycles is performed to reduce the random errors. Cyclic phenomena are clearly demonstrated by the post-processed data, and rapid void fraction change induced by the flashing is observed to be significant in the driving mechanism for low pressure oscillations. Parametric effects of inlet subcooling, heat flux, inlet flow restriction and system pressure are studied within the experiment range, and qualitative results are explained and compared with existing literature. The ratio between the oscillation period and the travelling time of the fluid particle through the test section is compared with the theoretical value for the density wave oscillation. The new dataset contains directly measured transient multi-dimensional two-phase data along the flow channel for natural circulation under density wave oscillations. The generated dataset is valuable for the validation of system analysis codes and CFD codes in the simulation of natural circulation phenomena and dynamic two-phase flow.

42 ENGINEERING↗

DOE OSTI2022

The role of cool-flame fluctuations in high-pressure spray flames, studied using high-speed optical diagnostics and Large-Eddy Simulations

Here this work investigates the low- and high-temperature ignition and combustion processes, applied to the Engine Combustion Network Spray A flame, combining advanced optical diagnostics and large-eddy simulations (LES). Simultaneous high-speed (50 kHz) formaldehyde (CH 2 0) planar laser-induced fluorescence (PLIF) and line-of-sight OH* chemiluminescence imaging were used to measure the low- and high-temperature flame, during ignition as well as during quasi-steady combustion. While tracking the cool flame at the laser sheet plane, the present experimental setup allows detection of distinct ignition spots and dynamic fluctuations of the lift-off length over time, which overcomes limitations for flame tracking when using schlieren imaging. After significant development to improve LES prediction of the low-and high-temperature flame position, both during the ignition processes and quasi-steady combustion, the simulations were analyzed to gain understanding of the mixture variance and how this variance affects formation/consumption of CH 2 0. Analysis of the high-temperature ignition period shows that a key improvement in the LES is the ability to predict heterogeneous ignition sites, not only in the head of the jet, but in shear layers at the jet edge close to the position where flame lift-off eventually stabilizes. The LES analysis also shows concentrated pockets of CH 2 0, in the center of jet and at 20 mm downstream of the injector (in regions where the equivalence ratio is greater than 6), that are of similar length scale and frequency as the experiment (approximately 5–6 kHz). The periodic oscillation of CH 2 0 match the frequency of pressure waves generated during auto-ignition and reflected within the constant-volume vessel throughout injection. The ability of LES to capture the periodic appearance and destruction of CH 2 0 is particularly important because these structures travel downstream and become rich premixed flames that affect soot production.

42 ENGINEERING↗

DOE OSTI2026

Measurement of the W-boson angular coefficients and transverse momentum in pp collisions at s=13 TeV with the ATLAS detector

The angular distributions of Drell–Yan lepton pairs provide sensitive probes of the underlying dynamics of quantum chromodynamics (QCD) effects in vector-boson production. This paper presents for the first time the measurement of the full set of angular coefficients together with the differential cross-section as a function of the transverse momentum of the W boson, in the full phase space of the decay leptons. The measurements are performed separately for the W-$$W^-$$ and W+$$W^+$$ channels. The analysis uses proton–proton collision data recorded by the ATLAS experiment at the Large Hadron Collider in 2017 and 2018, during special low-luminosity runs with a reduced number of interactions per bunch crossings (pile-up). The data correspond to an integrated luminosity of 338 pb-1$$^{-1}$$ at a centre-of-mass energy of s=13$$\sqrt{s} = 13$$ TeV. The low pile-up environment provides excellent experimental conditions for high-precision measurements of W-boson production. All results agree with theoretical predictions incorporating finite-order QCD corrections up to order αS2$$\alpha _S^2$$.

Aad, G↗

DOE OSTI2021

Hybrid Simulation Framework

HYBRID is a modeling toolset to assess the economic viability of Nuclear-Renewable Integrated Energy Systems (N-R IES). The frameworks enabling this toolset are INL’s RAVEN, its CashFlow plugin and the Modelica language. The toolset includes sample RAVEN workflows performing economic assessments. These workflows consist of: generation of stochastic time series and application of probabilistic analysis and optimization algorithms (RAVEN); a library of Modelica models representing the physical behavior of N-R IES; and the CashFlow plugin mapping physical performance to economic performance. The toolset allows assembling existing and new models such as nuclear reactors, renewable energy sources, energy storage, gas turbines, industrial processes, etc. into an N-R IES. The toolset workflows evaluate the dynamics of the N-R IES responding to stochastic conditions (electricity demand, price, etc.) and optimize the dispatch economics as well as N-R IES capacity planning.

Epiney, Aaron↗

DOE OSTI2022

Disruption Prediction and Avoidance in High Beta Long Pulse KSTAR Plasmas

This research is a concentrated and coordinated experimental and physics analysis program by a small U.S. team, focused on the topical area of “Long Pulse Control” as stated in the proposal solicitation. The overall goal of the research is to produce world-class analysis regarding forecasting and avoidance of disruptions in a long-pulse, superconducting tokamak at high stability parameters (e.g. normalized beta) with direct, quantifiable figures of merit (i.e. plasma disruption rate - “disruptivity”) and its reduction based on improved stability physics and control. The research targets the backbone of the problem – the basic, main issues – to produce this quantitative analysis by: (1) Creating and applying a new physics-based paradigm to understand how to eliminate disruptions: disruption event characterization, and event chain analysis – a direct manifestation of the approach suggested in the DOE Transients Report, (2) Ensuring that the components required to produce high-quality measurements and analysis needed for disruption event characterization for the tokamak in question (here, KSTAR) exist (e.g. pressure profile peaking, high-accuracy pitch angle data, compensated low frequency toroidal mode number n = 1 mode measurement) to create a primarily physics-based disruption forecasting capability, and (3) Addressing disruption avoidance for the first time in KSTAR by attacking primary disruption ‘event chains’, utilizing the most important associated plasma measurements (dynamic n = 1 field). The focus of this work will be avoidance of macroscopic instabilities, especially global modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Unified Language Frontend for Physic-Informed AI/ML

Artificial intelligence and machine learning (AI/ML) are becoming important tools for scientific modeling and simulation as in several other fields such as image analysis and natural language processing. ML techniques can leverage the computing power available in modern systems and reduce the human effort needed to configure experiments, interpret and visualize results, draw conclusions from huge quantities of raw data, and build surrogates for physics based models. Domain scientists in fields like fluid dynamics, microelectronics and chemistry can automate many of their most difficult and repetitive tasks or improve the design times by use of the faster ML-surrogates. However, modern ML and traditional scientific highperformance computing (HPC) tend to use completely different software ecosystems. While ML frameworks like PyTorch and TensorFlow provide Python APIs, most HPC applications and libraries are written in C++. Direct interoperability between the two languages is possible but is tedious and error-prone. In this work, we show that a compiler-based approach can bridge the gap between ML frameworks and scientific software with less developer effort and better efficiency. We use the MLIR (multi-level intermediate representation) ecosystem to compile a pre-trained convolutional neural network (CNN) in PyTorch to freestanding C++ source code in the Kokkos programming model. Kokkos is a programming model widely used in HPC to write portable, shared-memory parallel code that can natively target a variety of CPU and GPU architectures. Our compiler-generated source code can be directly integrated into any Kokkosbased application with no dependencies on Python or cross-language interfaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Integration of New Technology Considering the Trade-Offs Between Operational Benefits and Risks: A Case Study of Dynamic Line Rating

Electric grid operators are adept at handling complexity and uncertainty. However, with increasing introduction of renewable generation, distributed energy resources, and more frequent severe weather events, operators will experience new workload and challenging decision scenarios. Here, this paper quantifies risks and benefits from an operator's perspective of introducing weather based forecast Dynamic Line Ratings (DLR) using variable wind conditions in addition to ambient temperature to relieve transmission congestion and facilitating more offshore wind (OSW). A concept of operations (CONOPS) applied to a forecast DLR implementation and its integration with OSW is defined. A method for evaluating tradeoffs of derating to make the rating more conservative but decreasing the benefit was developed and applied to a case study for two existing overhead transmission lines on Long Island, New York. The CONOPS uses historical day-ahead and hour-ahead High Resolution Rapid Refresh weather forecasts and weather station data to support planning and real-time operations. The analysis determines the risk of downgrades in real-time operational rating compared to the forecast and quantifies the frequency and severity of last-minute downgrades. The risk is compared against the benefits in increased capacity to provide insights on the additional amount of uncertainty DLR and OSW will add to the operator's workload.

17 WIND ENERGY↗

DOE OSTI2020

Validation of finite element analysis strategy to investigate acoustic levitation in a two-axis acoustic levitator

A two-axis acoustic levitator can be used to generate a standing pressure wave capable of levitating solid and liquid particles at appropriate input conditions. This work proposes a simulation framework to investigate the two-axis levitation particle stability using a commercial, computational fluid dynamics software based on the harmonic solution to the acoustic wave equation. The simulation produced predictions of the standing wave that include a strong "+" shaped pattern of nodes and anti-nodes that are aligned with the levitator axes. To verify the simulation, a levitator was built and used to generate the standing wave. The field was probed with a microphone and a motorized-scanning system. After scaling the simulated pressure to the measured pressure, the magnitudes of the sound pressure level at corresponding high-pressure locations were different by no more than 5%. This is the first time a measurement of a two-axis levitator standing pressure wave has been presented and shown to verify simulations. Furthermore, as an additional verification, the authors consulted high speed camera measurements of a reference-levitator transducer, which was found to have a maximum peak-to-peak displacement of 50 +/- 5 mu m. The reference-levitator is known to levitate water at 160 dB. The system for this work was simulated to match the operation of the reference-levitator so that it produced sound pressure levels of 160 dB. This pressure was achieved when the transducer maximum peak-to-peak displacement was 50.8 mu m. The agreement between the two levitators' displacements provides good justification that the modeling approach presented here produces reliable results.

2 axis levitator↗

DOE OSTI2024

Market optimization and technoeconomic analysis of hydrogen-electricity coproduction systems

Decarbonization efforts across North America, Europe, and beyond rely on variable renewable energy sources such as wind and solar, as well as alternative fuels, such as hydrogen, to support the sustainable energy transition. These advancements have prompted a need for more flexibility in the electric grid to complement non-dispatchable energy sources and increased demand from electrification. Integrated energy systems are well suited to provide this flexibility, but conventional technoeconomic modeling paradigms neglect the time-varying dynamic nature of the grid and thus undervalue resource flexibility. In this work, we develop a computational optimization framework for dynamic market-based technoeconomic comparison of integrated energy systems that coproduce low-carbon electricity and hydrogen (e.g., solid oxide fuel cells, solid oxide electrolysis) against technologies that only produce electricity (e.g., natural gas combined cycle with carbon capture) or only produce hydrogen. Our framework starts with rigorous physics-based process models, built in the open-source Institute for the Design of Advanced Energy Systems (IDAES) modeling and optimization platform, for six energy process concepts. Using these rigorous models and a workflow to optimally design each technology, the framework is shown to be capable of evaluating new and emerging technologies in varying energy markets under a plethora of future scenarios (i.e., renewables penetration, carbon tax, etc.). Ultimately, our framework finds that solid oxide fuel cell-based coproduction systems achieve positive profits for 85% of the analyzed market scenarios. From these market optimization results, we use multivariate linear regression (R 2 values up to 0.99) to determine which electricity price statistics are most significant to predict the optimized annual profit of each system. The proposed framework provides a powerful tool for directly comparing flexible, multi-product energy process concepts to help discern optimal technology and integration options.

08 HYDROGEN↗

DOE OSTI2025

Structural Transformation and Degradation of Cu Oxide Nanocatalysts during Electrochemical CO 2 Reduction

The electrochemical CO 2 reduction reaction (CO 2 RR) holds enormous potential as a carbon-neutral route to the sustainable production of fuels and platform chemicals. The durability for long-term operation is currently inadequate for commercialization, however, and the underlying deactivation process remains elusive. A fundamental understanding of the degradation mechanism of electrocatalysts, which can dictate the overall device performance, is needed. In this work, we report the structural dynamics and degradation pathway of Cu oxide nanoparticles (CuO x NPs) during the CO 2 RR by using in situ small-angle X-ray scattering (SAXS) and X-ray absorption spectroscopy (XAS). The in situ SAXS reveals a reduction in the size of NPs when subjected to a potential at which no reaction products are detected. At potentials where the CO 2 RR starts to occur, CuO x NPs are agglomerated through a particle migration and coalescence process in the early stage of the reaction, followed by Ostwald ripening (OR) as the dominant degradation mechanism for the remainder of the reaction. As the applied potential becomes more negative, the OR process becomes more dominant, and for the most negative applied potential, OR dominates for the entire reaction time. The morphological changes are linked to a gradual decrease in the formation rate for multicarbon products (C 2 H 4 and ethanol). Other reaction parameters, including reaction intermediates and local high pH, induce changes in the agglomeration process and final morphology of the CuO x NPs electrode, supported by post-mortem ex situ microscopic analysis. The in situ XAS analysis suggests that the CuO x NPs reduced into the metallic state before the structural transformation was observed. The introduction of high surface area carbon supports with ionomer coating mitigates the degree of structural transformation and detachment of the CuO x NPs electrode. These findings show the dynamic nature of Cu nanocatalysts during the CO 2 RR and can serve as a rational guideline toward a stable catalyst system under electrochemical conditions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Coherences of Photoinduced Electron Spin Qubit Pair States in Photosystem I

Here, this publication presents the first comprehensive experimental study of electron spin coherences in photosynthetic reaction center proteins, specifically focusing on photosystem I (PSI). The ultrafast electron transfer in PSI generates spin-correlated radical pairs (SCRPs), which are entangled spin pairs formed in well-defined spin states (Bell states). Since their discovery in our group in the 1980s, SCRPs have been extensively used to enhance our understanding of structure–function relationships in photosynthetic proteins. More recently, SCRPs have been utilized as tools for quantum sensing. Electron spin decoherence poses a significant challenge in realizing practical applications of electron spin qubits, particularly the creation of quantum entanglement between multiple electron spins. This work is focused on the systematic characterization of decoherence in SCRPs of PSI. These decoherence times were measured as electron spin echo decay times, termed phase memory times (T M ), at various temperatures. Decoherence was recorded on both transient SCRP states P 700 + A 1 – and thermalized states. Our study reveals that T M exhibits minimal dependence on the biological species, biochemical treatment, and paramagnetic species. The analysis indicates that nuclear spin diffusion and instantaneous diffusion mechanisms alone cannot explain the observed decoherence. As a plausible explanation we discuss the assumption that the low-temperature dynamics of methyl groups in the protein surrounding the unpaired electron spin centers is the main factor governing the loss of the spin coherence in PSI.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Black hole perturbations and quasinormal modes in hybrid metric-Palatini gravity

The rapid advancement of gravitational wave astronomy in recent years has paved the way for the burgeoning development of black hole spectroscopy, which enhances the possibility of testing black holes by their quasinormal modes (QNMs). In this paper, the axial gravitational perturbations and the QNM frequencies of black holes in the hybrid metric-Palatini gravity (HMPG) are investigated. Additionally, the HMPG theory is characterized by a dynamical scalar degree of freedom and is able to explain the late-time accelerating expansion of the universe without introducing any ad hoc screening mechanism to preserve the dynamics at the Solar System scale. We obtain the master equation governing the axial gravitational perturbations of the HMPG black holes and calculate the QNM frequencies. Moreover, in the scrutiny of the black holes and their QNMs, we take into account the constraints on the model parameters based on the post-Newtonian analysis, and show how the QNM frequencies of the HMPG black holes would be altered in the observationally consistent range of parameter space.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Discovery of an unconventional quantum echo by interference of Higgs coherence

Detecting photon echoes from superconducting Higgs modes is challenging due to the necessity of preserving and retrieving phase coherence encoded in multiple Higgs and quasiparticle (QP) excitations. Here, we demonstrate the emergence of a Higgs echo in niobium superconductors. This approach disentangles unique quantum pathways involving the Higgs mode and QP excitations. Using Higgs echo spectroscopy, we also uncover unconventional echo formation caused by inhomogeneous broadening and “soft” QP bands, which dynamically evolve under terahertz (THz) driving. Specifically, THz pulse pairs modulate the superconducting gap, imprinting coherence and forming a temporal “Higgs grating.” This grating produces echoes with distinctive characteristics: (i) echo rephasing spectral peaks at superconducting gap frequencies, (ii) asymmetric echo formation delays unlike those observed in atoms or semiconductors, and (iii) negative-time echo signals stemming from Higgs-QP anharmonic interactions. Combined with advanced time-frequency analysis, these findings distinguish Higgs from QP responses and clarify their intricate interactions in THz-driven superconductivity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Electrochemical Residence Time Distribution as a Diagnostic Tool for Redox Flow Batteries

The fluid dynamic and electrochemical performance of redox flow batteries (RFBs) stems from the relationship between the flow field and the porous electrode, whose interplay determines how active species move and react during device operation. While characterization techniques, such as residence time distribution, offer insights into species mobility within a reactive volume for a traditional chemical reactor, electrochemical reactors also enable simultaneous measurement of the redox reactions, unlocking another dimension of analysis. Herein, we demonstrate how potentiodynamic measurements, using injections of electrolyte examined through moment analysis, can provide electrode-specific performance scaling relationships across a matrix of carbon paper and cloth electrodes with flow through and interdigitated flow fields. We further combine experimental campaigns with multiphysics simulations to demonstrate how electrode surface area can be estimated with this technique, which we then validate with activated and unactivated commercial carbon cloth electrodes. These studies reveal the multiscale observations that potentiodynamic measurements afford, augmenting existing electrochemical techniques for holistic electrochemical reactor diagnostics.

Electrochemistry↗

DOE OSTI2022

Materials Dynamics

The Materials Dynamics area of leadership focuses on understanding process-structure-properties-performance (PSPP) relationships for the extreme conditions of dynamic loading. This research encompasses controlled synthesis of materials to meet dynamic performance requirements and entails computational coupling across length and time scales for three-dimensional microstructure modeling. For this leadership area, we define dynamic loading as strain rates ≥ 10 3 /s and often, high pressures. A key grand challenge of this area is to predict and measure the evolution of microstructural phases, defect structures, and electronic structure under dynamic conditions while also measuring local temperature to understand transition states. Solving this challenge will require agile, multi-dimensional data analysis and interpretation capability.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Machine Learning for Anomaly Detection in Neural Network Security and SRF Cavities

This dissertation explores the development and deployment of machine learning approaches to address critical challenges in anomaly detection across two distinct domains: neural network security in federated learning settings and cavity behavior analysis in particle accelerator operations at Jefferson Lab in Newport News, Virginia. Anomaly detection identifies deviations from expected patterns, safeguarding systems in cybersecurity, industry, and research against malicious activities and failures. This dissertation demonstrates how our machine learning approaches enhance detection accuracy and efficiency in both neural network security and industrial applications. First, we investigate vulnerabilities in deep neural networks deployed in federated learning. Although federated learning preserves user privacy by training models locally, it remains vulnerable to backdoor attacks, in which malicious participants embed hidden triggers that induce targeted misbehavior. We propose a self-supervised contrastive learning framework to detect and mitigate such backdoor attacks. In our experiments, this method achieves higher detection accuracy and lower false positive rates than existing defenses, while operating without access to local model updates or original training data and thus preserving the privacy guarantees of the federated setting. Second, we address the operational reliability of superconducting radio-frequency (SRF) cavities at the Continuous Electron Beam Accelerator Facility (CEBAF). Our research leverages an unsupervised learning approach, combined with Principal Component Analysis (PCA) and k-means clustering, to identify anomalous behaviors in SRF cavities. Our method detects subtle anomalous behavior by analyzing SRF signal data. This knowledge allows for the early detection and resolution of potential faults, significantly improving the efficiency and reliability of operations. Third, we extend these insights to time-series anomaly detection more broadly. We design a contrastive-learning based model tailored to increasingly dynamic environments and academic research. This model improves detection accuracy in settings that require real-time monitoring and predictive maintenance. Our research underscores the broader applicability and impact of advanced machine learning techniques in anomaly detection. By extracting meaningful patterns from complex data, machine learning can significantly enhance security in distributed neural networks and improve the efficiency of particle accelerator operations. This dissertation serves as a stepping stone for future investigations into the vast possibilities of anomaly detection, inspiring further exploration and development of machine learning techniques in this field.

Ferguson, Hal [Old Dominion University]↗

DOE OSTI2021

Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Lastly, potential applications and caveats to the usage of LD are included as a summary.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗