Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “operator”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A comprehensive and fair comparison of two neural operators (with practical extensions) based on $\mathrm{FAIR}$ data

Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applications as well as for system identification in science and engineering. Herein, we investigate the performance of two neural operators, which have shown promising results so far, and we develop new practical extensions that will make them more accurate and robust and importantly more suitable for industrial-complexity applications. The first neural operator, DeepONet, was published in 2019 (Lu et al., 2019), and its original architecture was based on the universal approximation theorem of Chen & Chen (1995). The second one, named Fourier Neural Operator or FNO, was published in 2020, and it is based on parameterizing the integral kernel in the Fourier space. DeepONet is represented by a summation of products of neural networks (NNs), corresponding to the branch NN for the input function and the trunk NN for the output function; both NNs are general architectures, e.g., the branch NN can be replaced with a CNN or a ResNet. According to Kovachki et al. (2021), FNO in its continuous form can be viewed conceptually as a DeepONet with a specific architecture of the branch NN and a trunk NN represented by a trigonometric basis. In order to compare FNO with DeepONet computationally for realistic setups, we develop several extensions of FNO that can deal with complex geometric domains as well as mappings where the input and output function spaces are of different dimensions. We also develop an extended DeepONet with special features that provide inductive bias and accelerate training, and we present a faster implementation of DeepONet with cost comparable to the computational cost of FNO, which is based on the Fast Fourier Transform. Here we consider 16 different benchmarks to demonstrate the relative performance of the two neural operators, including instability wave analysis in hypersonic boundary layers, prediction of the vorticity field of a flapping airfoil, porous media simulations in complex-geometry domains, etc. We follow the guiding principles of FAIR (Findability, Accessibility, Interoperability, and Reusability) for scientific data management and stewardship. The performance of DeepONet and FNO is comparable for relatively simple settings, but for complex geometries the performance of FNO deteriorates greatly. We also compare theoretically the two neural operators and obtain similar error estimates for DeepONet and FNO under the same regularity assumptions.

42 ENGINEERING↗
DOE OSTI2025

A higher-order finite-element implementation of the nonlinear Fokker–Planck collision operator for charged particle collisions in a low density plasma

Collisions between particles in a low density plasma are described by the Fokker–Planck collision operator. In applications, this nonlinear integro-differential operator is often approximated by linearised or ad-hoc model operators due to computational cost and complexity. In this work, we present an implementation of the nonlinear Fokker–Planck collision operator written in terms of Rosenbluth potentials in the Rosenbluth–MacDonald–Judd (RMJ) form. The Rosenbluth potentials may be obtained either by direct integration or by solving partial differential equations (PDEs) similar to Poisson's equation: we optimise for performance and scalability by using sparse matrices to solve the relevant PDEs. We represent the distribution function using a tensor-product continuous-Galerkin finite-element representation and we derive and describe the implementation of the weak form of the collision operator. We present tests demonstrating a successful implementation using an explicit time integrator and we comment on the speed and accuracy of the operator. Finally, we speculate on the potential for applications in the current and next generation of kinetic plasma models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Strategies for microgrid operation under real-world conditions

Microgrids are an increasingly relevant technology for integrating renewable energy sources into electricity systems. Based on a microgrid implementation in California, in this study we investigate microgrid operation under real-world conditions. These conditions have not yet been considered in combination and encompass energy charges, demand charges, export limits, as well as uncertainty about future electricity demand and generation in the microgrid. Under these conditions, we evaluate the performance of two frequently applied groups of strategies for microgrid operation. The first group is composed of proactive strategies that optimize decisions based on forecasts of future electricity generation and demand. The second group includes reactive strategies that make operational decisions based exclusively on the current state of the microgrid. We evaluate the performance of the strategies under varying operational parameters, forecast accuracies, and microgrid configurations—well beyond our Californian showcase. Our results confirm the expectation that proactive strategies outperform reactive ones in the majority of settings. Yet, reactive strategies can perform better under short control intervals or under moderate prediction errors of PV generation or demand. Furthermore, the interplay between real-world conditions and operational strategies reveals several additional insights for research on microgrid operation. First, we find that demand charges and export limits decisively affect microgrid performance. Second, the impact of forecast errors is highly non-linear and non-monotonous. Third, escalating negative interactions between forecast errors and demand charges make proactive strategies benefit from longer control intervals. This result is contrary to existing best practice, which promotes short control intervals to minimize the impact of uncertainty.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Co-optimization of nuclear reactor flexible power operation and maintenance scheduling

As flexible power operation of nuclear power plants becomes more attractive due to the reduction in fossil-fueled dispatchable generation on energy grids, finding optimal power production strategies that balance revenue generation with operational concerns becomes more complex. This article presents a general framework to aid operators in designing economically optimal long term dispatch strategies for nuclear power plants. The principal novelty is the linking of estimated system remaining useable life (RUL) to strategic operational decisions. It is shown that, depending on the relationship between the fixed costs from maintenance and the associated lost revenue from an outage, it can be economically optimal in the long term to delay a maintenance outage and not perform this alongside refueling. For a given relationship between power ramping and degradation, optimal strategies were found that discouraged load following in some situations while minimizing unnecessary maintenance. It is shown that heavy load following can cause maintenance and refueling outages to diverge due to their inverse relationships with respect to load following, potentially leading to a significant loss in capacity factor. As a result, this general framework can be applied to specific reactor dispatch allowing operators to adapt operational strategies as future grid conditions change.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2022

Neural-network based collision operators for the Boltzmann equation

Kinetic gas dynamics in rarefied and moderate-density regimes have complex behavior associated with collisional processes. These processes are generally defined by convolution integrals over a high-dimensional space (as in the Boltzmann operator), or require evaluating complex auxiliary variables (as in Rosenbluth potentials in Fokker-Planck operators) that are challenging to implement and computationally expensive to evaluate. In this work, we develop a data-driven neural network model that augments a simple and inexpensive BGK collision operator with a machine-learned correction term, which improves the fidelity of the simple operator with a small overhead to overall runtime. The composite collision operator has a tunable fidelity and, in this work, is trained using and tested against a direct-simulation Monte-Carlo (DSMC) collision operator.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Assessment of small mechanical wastewater treatment plants: Relative life cycle environmental impacts of construction and operations

Many slow growing and shrinking rural communities struggle with aging or inadequate wastewater treatment plants (WWTPs), and face challenges in constructing and operating such facilities. Although existing literature has provided insight into the environmental sustainability of large facilities, including both the construction and operational phases, these studies have not examined small, rural facilities treating less than 7,000 m 3 /d (1.8 MGD) of wastewater in adequate depth and breadth. In this study, a detailed inventory of the construction and operational data for 16 case studies of small WWTPs was developed to elucidate their environmental life cycle impacts. Conventional LCA framework was followed. The results show that the environmental impacts of both the construction and operational phases are considerable. Operational impacts are highly related to energy usage. Improving energy efficiency of a plant may reduce the environmental impacts related to operations. Construction impacts can vary considerably between facilities. Process-related factors (e.g., concrete and reinforcing steel used in basins) are typically sized using the design flow; thus much of the variability in construction impacts among plants stems from the non-process related infrastructure. Multiple regression analysis was used as an exploratory tool to identify which non-process related plant aspects contribute to the variable environmental impact of small WWTPs. These factors include aluminum, cast iron, and the capacity utilization ratio (defined as the ratio of average flow to design flow). Furthermore, industry practitioners should consider these factors when aiming to reduce the environmental impacts of a small WWTP related to construction. Scenario sensitivity analyses found that the environmental impact of construction became smaller with longer design life, and the end-of-life consideration does not heavily influence the environmental sustainability of a WWTP.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2023

Creating an Advanced Sensor Network to calculate real-time, mass-weighted flue gas composition and air heater leakage of a coal-fired utility boiler under dynamic operating conditions

Utilization of renewable energy sources to minimize the environmental impact of energy production has changed the way utility boilers operate, requiring frequent load cycling between full load and partial loads as low as 30%. Dynamic operation of coal-fired utility boilers significantly reduces boiler efficiency when compared to steady state at full load. Data-driven plant optimization has shown success with coal-fired utility boilers under dynamic operating conditions. The purpose of this work was to create an Advanced Sensor Network (ASN) to provide more extensive real-time data to inform dynamic plant optimization of Net Unit Heat Rate (NUHR). The ASN consists of gas sampling grids in the convective pass of the boiler and downstream of the air heater. These sampling grids allow for quantification of spatial variation of flue gas within the boiler and calculation of mass-weighted composition of flue gas through the combination of composition, velocity, and temperature measurements. The comparison of O 2 between the inlet and outlet of the air heater is used to calculate air leakage in real time. Flue gas composition and air heater leakage are both important factors in boiler efficiency and NUHR. Further, the results of this work support the value of mass-weighted averages for determining flue gas composition accurately. The measurements from the ASN show increased composition stratification during dynamic operation, with an average standard deviation 38% higher than observed during steady-state operation. Air heater leakage was also observed to increase from 2.8% to 5.1% following a load change. Prior to the installation of the ASN, these data would not have been available for dynamic control. These real-time data will be leveraged to calculate and optimize for NUHR during dynamic operation in future work.

42 ENGINEERING↗
DOE OSTI2020

Assessment of operational availability for the PIP-II Superconducting Radio Frequency linear accelerator facility

Operational availability is a critical performance measure for an accelerator facility in modern time. A high availability enables the facility to serve a wide range of users simultaneously. Consequently, besides pure accelerator physics considerations, newly proposed accelerator facilities account for the availability and reliability aspects in the design phases. It allows incorporation of appropriate mitigation strategies for the most vulnerable systems in the machine and therefore, minimizes unscheduled interruptions during the operation. This report lays out a methodology for the availability assessment of the complete particle accelerator facility and presents an initial assessment of the availability of the newly proposed Proton Improvement Plan-II (PIP-II) accelerator facility at Fermilab. The paper describes a comprehensive reliability model of the PIP-II facility that comprises not only 800 MeV linear accelerator (linac) system but also essential utility systems in the form of cryogenic, water, power and air systems. The paper details estimations of the availability of the PIP-II facility for two operational modes i.e. the nominal operational mode featuring 800 MeV beam and critical operational mode involving operation with the lowest objective beam energy of 600 MeV.

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Data-driven linear time advance operators for the acceleration of plasma physics simulation

In this study, we demonstrate the application of data-driven linear operator construction for time advance with a goal of accelerating plasma physics simulation. We apply dynamic mode decomposition (DMD) to data produced by the nonlinear SOLPS-ITER (Scrape-off Layer Plasma Simulator - International Thermonuclear Experimental Reactor) plasma boundary code suite in order to estimate a series of linear operators and monitor their predictive accuracy via online error analysis. We find that this approach defines when these dynamics can be represented by a sequence of approximate linear operators and is essential for providing consistent projections when compared to an unconstrained application. For linear diffusion and advection–diffusion fluid test problems, we construct and apply operators within explicit and implicit time advance schemes, demonstrating that stability can be robustly guaranteed in each case. We further investigate the use of the linear time advance operators within several integration methods including forward Euler, backward Euler, and the matrix exponential. The application of this method to simulation data from SOLPS-ITER, with varying levels of Markov chain Monte Carlo numerical noise, shows that constrained DMD operators yield a capability to identify, extract, and integrate a (slow) subset of the present timescales. Example applications show that for projected speedup factors of [Formula: see text], and [Formula: see text], a mean relative error of 3%, 5%, and 8% and maximum relative error less than 20% are achievable, which appears acceptable for typical SOLPS-ITER steady-state simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

PFPO plasma scenarios for exploration of long pulse operation in ITER

Long Pulse Scenarios (LPS) in ITER foreseen during the Pre-Fusion Power Operation (PFPO) phase of the ITER Research Plan (IRP) are assessed using 1.5D transport simulations within the ASTRA framework. Such assessment is required to predict the operational space for LPS operation in PFPO, as well as to evaluate which physics processes for LPS operation during Fusion Power Operation (FPO) could be studied during PFPO. An important aspect in the development of LPSs in PFPO is to minimize lifetime consumption of the Central Solenoid (CS) for these scenarios. The maximum pulse length achievable for LPSs in PFPO with no consumption of CS lifetime (currents in CS coils $⩽$30 kA per turn) has been assessed for a range of heating schemes and heating mixes, confinement regimes (L-mode and H-mode) and for helium and hydrogen plasmas. The operational space of LPS and pulse length has been explored through density scans with the Heating and Current Drive mix required for the FPO Q $⩾$ 5 steady-state plasma scenario (namely Neutral Beam Injection and Electron Cyclotron Heating) including acceptable shine through losses on the first wall for both helium and hydrogen plasmas. Fast particle physics aspects that are common between FPO plasmas and LPS PFPO H-mode plasmas at low densities are studied including MHD stability analysis with the KINX code and non-perturbative critical gradient model based on high-n Toroidal Alfven Eigenmodes (TAE) stability kinetic ballooning code HINST calculations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Pre- and post-launch operation of the Resolve soft X-ray spectrometer onboard the XRISM satellite

Resolve is a high-resolution X-ray spectrometer onboard the X-Ray Imaging and Spectroscopy Mission (XRISM), launched on September 6 (UT), 2023. The Resolve has performed better than its required spectral resolution (7 eV at full width at half maximum at 6 keV), both on the ground and in orbit, and has been confirmed to have comparable performance to the soft X-ray spectrometer onboard the ASTRO-H (Hitomi) satellite. The focal plane is composed of an array of microcalorimeter detectors operated at 50 mK to achieve the required energy resolution, and the cooling system is designed to satisfy the lifetime requirement of over 3 years. The focal plane and cooling system are contained in a vacuum-insulated dewar. The cooling system is equipped with a two-stage adiabatic demagnetization refrigerator (ADR) that uses superfluid liquid helium (LHe) as its heat sink. The system includes a third ADR stage that can be used to provide the heat sink when the helium is exhausted. A Joule–Thomson cooler and several two-stage Stirling coolers are used to reduce the heat load on the LHe. During pre-launch operations, we carried out a superfluid LHe top-off operation. The resultant amount of LHe onboard Resolve was over 35 L before launch, which is sufficient to meet the lifetime requirement. During post-launch operation, the LHe vent valve was opened 5 min after launch during rocket acceleration, and the cryocoolers were turned on after several orbits, as planned, which established stable cooling within the dewar. Pre- and post-launch operations for the Resolve instrument were planned around multiple constraints from launch vehicle operations; all were successfully completed, and the launch requirements were fully met.

X-Ray Imaging and Spectroscopy Mission↗
DOE OSTI2020

Report on the Use and Function of the Integrated Operations Capability Analysis Platform (ICAP) and the LWRS Innovation Portal (IP)

The commercial nuclear power industry has achieved excellent safety and reliability performance but is struggling to survive in an electricity market that is increasingly dominated by subsidized renewables and cheap natural gas. These challenges have forced utilities to explore previously uncharted avenues to drastically reduce the operations and maintenance costs of their plants which are the primary drivers of the total cost to produce electricity. The petrochemical industry faced similar challenges some years ago as the costs of extraction and processing were rising while their reservoirs were being depleted along with a drop-in commodity prices that resulted in unsustainable operations. In this challenging climate, they developed a business model called Integrated Operations (IO) that sought to utilize technology to enable news ways of working through the integration of people, technology, process and governance changes. The Light Water Reactor Sustainability (LWRS) program, working with IFE have developed an operating model via transferable learnings from the North Sea O&G industry. This framework is termed “Integrated Operations for Nuclear” (ION). ION is a transformation model that integrates the benefits and features of four principal factors: People, Technology, Process and Governance. The purpose of this report is to describe how to generate an ION business process analysis and utilize this information to reduce O&M costs. In order to make this job easier, DOE LWRS has created a suite of tools that will allow a person who is involved in a transformation effort at their company to build a solid documented business case for embarking on a major transformation effort. These tools, the Integrated Operations Capability Analysis Model (ICAP) and the LWRS Innovation Portal (IP) are described herein with instructions on their use. Instructions on how the interface with the EPRI Business Case Analysis Method (BCAM) are also provided.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2024

Solar and Storage Integration in the Southeastern United States: Economics, Reliability, and Operations

Solar energy has the potential to be a core energy resource for the southeastern United States. To better understand the implications of higher levels of solar PV (27%-43% of total generation capacity) and electricity storage (13%-49% of peak load) would affect electricity system reliability, costs, and operations in the U.S. Southeast, this study sought to address two main questions. First, how would higher levels of solar PV and electricity storage impact the costs, reliability, and operations of electricity systems in the Southeast in 2035? Second, at different levels of solar PV and electricity storage, what are the benefits of operational coordination among utilities in the Southeast, through more efficient regional dispatch and sharing operating reserves? To answer these questions, the study used detailed capacity expansion and dispatch modeling to develop and examine 15 scenarios with different levels of solar PV, electricity storage, and operational coordination, focusing on the year 2035. The study also evaluates the benefits of operational coordination among utilities through more efficient regional dispatch and reserve sharing, at different levels of solar and storage. The study focuses on five balancing regions that cover Alabama, Georgia, Kentucky, North Carolina, South Carolina, Tennessee, and parts of Mississippi and Missouri.

14 SOLAR ENERGY↗
DOE OSTI2021

Electrification Futures Study: Power Systems Operation with Newly Electrified and Flexible Loads

The Electrification Futures Study (EFS) is a multi-year study designed to analyze the impacts of widespread electrification in the U.S. energy system. In this webinar we present the detailed power systems operational analysis from the final report in the EFS series. For this analysis, we simulate multiple 2050 power systems for the conterminous United States to assess how variations in the magnitude and shape of electricity demand driven by electrification, and the extent of load participation to more-actively provide grid services, might impact the hourly operation, operational costs, and emissions of various power systems in 2050. The impacts of electrification and demand flexibility are overlayed across systems with significantly greater penetrations of variable renewable energy than today. Overall, we find that the high electrification scenarios envisioned in the EFS with significant VRE penetration (66% of annual national generation) can be operated to meet future increased levels of electrified demand. We also find that demand-side flexibility can enhance operational efficiency and reduce overall annual production costs by $5-$10 billion. The complementary relationship between flexible electric vehicle charging loads and solar generation is particularly pronounced, but in the absence of demand-side flexibility electrification can lead to increased wind and solar curtailment. By helping to reduce renewable curtailment, flexibility can also reduce power sector CO2 emissions. The analysis highlights the value of increased integration and coordination of demand- and supply-side resources in future electric system planning and operations—particularly under high electrification futures.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

Enhancing Biomass Productivity by Forecast‐Informed Pond Operations

ABSTRACT Microalgal cultivation for biofuels and proteins holds significant promise but faces challenges in achieving economically viable biomass productivity under variable environmental conditions. This study introduces a forecast‐informed pond operation (FIPO) system that uses numerical weather prediction (NWP) ensemble forecasts and the biomass assessment tool (BAT) to optimize daily dilution rates for enhanced biomass production. In contrast to the current practice, where fixed dilution rates are based on operator experience, the FIPO system determines the optimal dilution rate based on future weather forecasts and biomass growth conditions. Our experiments validate the effectiveness of FIPO in both short‐ and long‐term growth scenarios. In short‐term experiments, FIPO increased biomass production by 21.3% compared to batch growth and 7.4% over fixed dilution (60% every 3 days) operations. The NWP forecast‐informed operations achieved biomass production nearly identical to that using perfect weather forecasts, highlighting the accuracy of current NWP forecasts for guiding pond operations. In long‐term experiments, FIPO resulted in biomass production increases of 13.3% and 17.8% compared to two fixed dilution rates (60% every 3 days and 20% daily). These findings underscore the viability of using NWP forecasts to optimize microalgal cultivation systems. By adjusting daily dilution rates in response to forecasted weather, operators can achieve higher biomass yields and mitigate risks associated with environmental variability. This study provides a foundation for future research and practical applications in commercial‐scale microalgal production.

Yan, Hongxiang [Energy and Environment Directorate↗
DOE OSTI2023

Local element operations for curved simplex meshes

Summary Mesh optimization procedures are generally a combination of node smoothing and discrete operations which affect a small number of elements to improve the quality of the overall mesh. These procedures are useful as a post‐processing step in mesh generation procedures and in applications such as fluid simulations with severely deforming domains. In order to perform high‐order mesh optimization, these ingredients must also be extended to high‐order (curved) meshes. In this work, we present a method to perform local element operations on curved meshes. The mesh operations discussed in this work are edge/face swaps, edge collapses, and edge splitting (more generally refinement) for triangular and tetrahedral meshes. These local operations are performed by first identifying the patch of elements which contain the edge/face being acted on, performing the operation as a “straight‐sided one" by placing the high‐order nodes via an isoparametric mapping from the master element, and smoothing the high‐order nodes on the elements in the patch by minimizing a Jacobian‐based high‐order mesh distortion measure. Since the initial “straight‐sided guess” from the placement of the nodes via the isoparametric mapping frequently results in invalid elements, the distortion measure must be regularized which allows for mesh untangling for the optimization to succeed. We present several examples in 2D and 3D to demonstrate these local operations and how they can be combined with a high‐order node smoothing procedure to maintain mesh quality when faced with severe deformations.

Shi, A.↗
DOE OSTI2022

The Branes Behind Generalized Symmetry Operators

The modern approach to m-form global symmetries in a d-dimensional quantum field theory (QFT) entails specifying dimension d–m–1 topological generalized symmetry operators which non-trivially link with m-dimensional defect operators. In QFTs engineered via string constructions on a non-compact geometry X, these defects descend from branes wrapped on non-compact cycles which extend from a localized source / singularity to the boundary ∂X. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in ∂X. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F-theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3-form symmetry detect a non-invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗