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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
NASA NTRS2000

Computations of Boiling in Microgravity

The absence (or reduction) of gravity, can lead to major changes in boiling heat transfer. On Earth, convection has a major effect on the heat distribution ahead of an evaporation front, and buoyancy determines the motion of the growing bubbles. In microgravity, convection and buoyancy are absent or greatly reduced and the dynamics of the growing vapor bubbles can change in a fundamental way. In particular, the lack of redistribution of heat can lead to a large superheat and explosive growth of bubbles once they form. While considerable efforts have been devoted to examining boiling experimentally, including the effect of microgravity, theoretical and computational work have been limited. Here, the growth of boiling bubbles is studied by direct numerical simulations where the flow field is fully resolved and the effects of inertia, viscosity, surface deformation, heat conduction and convection, as well as the phase change, are fully accounted for. Boiling involves both fluid flow and heat transfer and thus requires the solution of the Navier-Stokes and the energy equations. The numerical method is based on writing one set of governing transport equations which is valid in both the liquid and vapor phases. This local, single-field formulation incorporates the effect of the interface in the governing equations as source terms acting only at the interface. These sources account for surface tension and latent heat in the equations for conservation of momentum and energy as well as mass transfer across the interface due to phase change. The single-field formulation naturally incorporates the correct mass, momentum and energy balances across the interface. Integration of the conservation equations across the interface directly yields the jump conditions derived in the local instant formulation for two-phase systems. In the numerical implementation, the conservation equations for the whole computational domain (both vapor and liquid) are solved using a stationary grid and the phase boundary is followed by a moving unstructured two-dimensional grid. While two-dimensional simulations have been used for preliminary studies and to examine the resolution requirement, the focus is on fully three-dimensional simulations. The numerical methodology, including the parallelization and grid refinement strategy is discussed, and preliminary results shown. For buoyancy driven flow, the heat transfer is in good agreement with experimental correlations. The changes when gravity is turned off and/or fluid shear is added are discussed, as well as the difference between simulations of a layer freely releasing bubbles versus simulations using only one wavelength initial perturbation. Figure 1 shows the early stages of the formation of a three-dimensional bubble from a thin vapor layer. The boundary conditions are periodic in the x and y direction, the bottom is a hot and the top allows a free outflow. The jagged edge of the surface close to the bottom of the computational domain is due to some of the surface elements being on the other side of the domain and some elements not plotted by our plotting routine. In the second figure, we show the temperature distribution through two perpendicular planes.

Tryggvason, G.↗
DOE OSTI2023

LANL Institutional Computing Close-out Report for Project t22_ocean_time_step

This year, my team made a very productive use of LANL IC time. Four papers were published that used IC resources, and two more are under review. These publications fall into three categories: 1.) Improving tide modeling in the global ocean by adding self attraction and loading (Barton et. al. 2022), ice shelf cavities (Pal et al. 2023), and local time stepping (Lilly et al. 2023). 2.) A new sea ice numerical formulation (Capodaglio 2023). 3.) Performance comparisons, methods, and test cases for ocean model development, This includes a verification suite for ocean models (Bishnu, submitted) and a comparison between Julia and Fortran (Strauss, submitted). A new time-stepping method was introduced in Calandrini et al. (2022). These publications, and the use of LANL IC resources, go hand-in-hand with our mentoring efforts to train young scientists. Two of the lead authors are graduate students who are conducting their PhD research: Kristin Barton at the University of Michigan and Jeremy Lilly at Oregon State. These are both their first publications, and they both have DOE funding and DOE mentors. In addition, Bishnu is a post-doctoral researcher at LANL; Strauss conducted his research as a senior in high school; and Capodaglio, Calandrini and Pal are all early-career scientists who were converted to staff in 2021 or 2022. Here we highlight two publications on improvements in tidal modeling: Barton et al. 2022 and Lilly et al. 2023.

58 GEOSCIENCES↗
NASA NTRSDate not supplied

A Scenario-Based Approach to Assess Continuity Gaps in Earth Observations

Decision analysis processes outlined in systems engineering references, such as the Systems Engineering Body of Knowledge [1] or the NASA Systems Engineering Handbook [2], recommend following a series of steps to support decision-making for engineering applications. These steps are consistent across the literature and typically involve defining objectives, defining relevant criteria against which candidate alternatives can be assessed, selecting an evaluation method, assessing alternatives, and making a recommendation based on this assessment. These approaches are well-suited to assess systems for which sets of common objectives and constraints can be identified. A vast body of literature describes their application to numerous engineering problems, and many methods have been developed since the middle of the 20th century to support the assessment of candidate alternatives for such systems [3, 4, 5, 6]. These approaches, however, show some limitations when decisions pertain to systems of systems. These collections of individual systems typically do not share a common set of objectives and constraints, tend to be highly complex, and the definition of assessable, high-value candidate alternatives poses a challenge due to interdependencies. The formulation, development, operation, and funding of Earth observing missions are such that gaps may occur between missions, therefore impacting the continuity of measurements. These Earth observing mission architectures constitute systems of systems for which a common set of objectives and constraints can be challenging to define. When tasked with assessing gaps that may occur for a series of spaceborne missions that address similar science parameters, the authors therefore proposed to depart from the traditional objective- and criteria-based approach and instead adopted a scenario-based approach. This approach does not require a prioritized set of common objectives; rather, it assesses the impact of possible decisions on the system-of-systems and characterizes its possible future states. It also offers the flexibility to adjust assumptions, update inputs, and refine supporting models over time, while enabling the rapid, early identification of challenges and key decision points. By simulating the impact of potential decisions on the entirety of the system-of-systems, the approach enables the synchronization of multiple decisions that typically occur at the level of the individual system. This paper provides an overview of the approach that was developed to assess continuity gaps for a series of operational and planned spaceborne Earth observing missions, a description of its application to the problem at hand, and a discussion of currently known limitations.

scenario earth observations continuity gaps↗
NASA NTRSDate not supplied

A Scenario-Based Approach to Assess Continuity Gaps in Earth Observations

Decision analysis processes outlined in systems engineering references, such as the Systems Engineering Body of Knowledge [1] or the NASA Systems Engineering Handbook [2], recommend following a series of steps to support decision-making for engineering applications. These steps are consistent across the literature and typically involve defining objectives, defining relevant criteria against which candidate alternatives can be assessed, selecting an evaluation method, assessing alternatives, and making a recommendation based on this assessment. These approaches are well-suited to assess systems for which sets of common objectives and constraints can be identified. A vast body of literature describes their application to numerous engineering problems, and many methods have been developed since the middle of the 20th century to support the assessment of candidate alternatives for such systems [3, 4, 5, 6]. These approaches, however, show some limitations when decisions pertain to systems of systems. These collections of individual systems typically do not share a common set of objectives and constraints, tend to be highly complex, and the definition of assessable, high-value candidate alternatives poses a challenge due to interdependencies. The formulation, development, operation, and funding of Earth observing missions are such that gaps may occur between missions, therefore impacting the continuity of measurements. These Earth observing mission architectures constitute systems of systems for which a common set of objectives and constraints can be challenging to define. When tasked with assessing gaps that may occur for a series of spaceborne missions that address similar science parameters, the authors therefore proposed to depart from the traditional objective- and criteria-based approach and instead adopted a scenario-based approach. This approach does not require a prioritized set of common objectives; rather, it assesses the impact of possible decisions on the system-of-systems and characterizes its possible future states. It also offers the flexibility to adjust assumptions, update inputs, and refine supporting models over time, while enabling the rapid, early identification of challenges and key decision points. By simulating the impact of potential decisions on the entirety of the system-of-systems, the approach enables the synchronization of multiple decisions that typically occur at the level of the individual system. This paper provides an overview of the approach that was developed to assess continuity gaps for a series of operational and planned spaceborne Earth observing missions, a description of its application to the problem at hand, and a discussion of currently known limitations.

scenario earth observations continuity gaps↗
NASA NTRS1997

Structural Verification of the Space Shuttle's External Tank Super LightWeight Design: A Lesson in Innovation

The Super LightWeight Tank (SLWT) team was tasked with a daunting challenge from the outset: boost the payload capability of the Shuttle System by safely removing 7500 lbs. from the existing 65,400 lb. External Tank (ET). Tools they had to work with included a promising new Aluminum Lithium alloy, the concept of a more efficient structural configuration for the Liquid Hydrogen (LH2) tank, and a highly successful, mature Light Weight Tank (LWT) program. The 44 month schedule which the SLWT team was given for the task was ambitious by any measure. During this time the team had to not only design, build, and verify the new tank, but they also had to move a material from the early stages of development to maturity. The aluminum lithium alloy showed great promise, with an approximately 29% increase in yield strength, 15% increase in ultimate strength, 5 deg/O increase in modulus and 5 deg/O decrease in density when compared to the current 2219 alloy. But processes had to be developed and brought under control, manufacturing techniques perfected, properties characterized, and design allowable generated. Because of the schedule constraint, this material development activity had to occur in parallel with design and manufacturing. Initial design was performed using design allowable believed to be achievable with the Aluminum Lithium alloy system, but based on limited test data. Preliminary structural development tests were performed with material still in the process of iteration. This parallel path approach posed obvious challenges and risks, but also allowed a unique opportunity for interaction between the structures and materials disciplines in the formulation of the material.

Otte, Neil↗
DOE OSTI2025

Marine Hydrogen Demonstration

This report summarizes Phase 1 of a project involving the design of a Floating Hydrogen Production and Dispensing Barge destined for the Port of San Francisco (SF). The H 2 Barge is designed to produce renewable H 2 at the rate of ~ 530 kg/day, storing 512 kg of hydrogen at 517-bar, allowing fast refueling of hydrogen fuel cell vessels and land-side hydrogen delivery trailers for distribution into the nascent SF hydrogen ecosystem. The broader considerations that impacted the H2 Barge design are also described. An account is given of a new review process formulated by the United States Coast Guard (USCG) to review this first-of-its-kind maritime implementation of hydrogen technology. The immediate goals of the H 2 Barge Project are to 1) demonstrate the feasibility, viability and methods of hydrogen production, storage and fueling in a maritime context, 2) help shape (where needed) and navigate the required local, state and federal regulatory gauntlet and 3) catalyze a “green hydrogen ecosystem” (both marine and landside) with locally produced renewable hydrogen at the San Francisco waterfront. A summary is also given of the modeling and experimental activity of Phase 1 directed to the development of science-based refueling protocols for large marine Type IV 250-bar hydrogen tanks. Combined modeling and experimental studies are reported of the filling of large (28 kg) 250-bar Type IV hydrogen tanks of the type being deployed on early hydrogen ferries, such as the MV Sea Change. The primary question was to determine how such tanks can be successfully filled (state of charge greater than 97%) within 45 minutes without exceeding the 82 °C temperature limit historically set for such tanks. The studies show that a gas injector is needed avoid thermal stratification during filling which can result in potential hot spots. Pre-cooling of the hydrogen was found to be essential in most cases, as ambient conditions greatly affect the need for a pre-cooling to achieve the 45-minute fill time. Pre-cooling cannot be supplied by nearby water, such as that found in nature (bays, lakes, rivers, etc.) because pre-cooling cooling below 0 °C was found to be necessary to avoid excessive compression heating. The experimental results afforded a calibration of the engineering model (SOFIL) for these large 250-bar tanks, which now enables using SOFIL to predict volume-averaged hydrogen filling temperatures to an accuracy of +/- 2.7°C for these tanks. The model can therefore be used to evaluate potential scenarios for development of a standardized fueling methodology for ferries utilizing large Type-IV tanks of the type examined here.

08 HYDROGEN↗
NASA NTRS1990

Acoustic design criteria in a general system for structural optimization

Passenger comfort is of great importance in most transport vehicles. For instance, in the new generation of regional turboprop aircraft, a low noise level is vital to be competitive on the market. The possibilities to predict noise levels analytically has improved rapidly in recent years. This will make it possible to take acoustic design criteria into account in early project stages. The development of the ASKA FE-system to include also acoustic analysis has been carried out at Saab Aircraft Division and the Aeronautical Research Institute of Sweden in a joint project. New finite elements have been developed to model the free fluid, porous damping materials, and the interaction between the fluid and structural degrees of freedom. The FE approach to the acoustic analysis is best suited for lower frequencies up to a few hundred Hz. For accurate analysis of interior cabin noise, large 3-D FE-models are built, but 2-D models are also considered to be useful for parametric studies and optimization. The interest is here focused on the introduction of an acoustic design criteria in the general structural optimization system OPTSYS available at the Saab Aircraft Division. The first implementation addresses a somewhat limited class of problems. The problems solved are formulated: Minimize the structural weight by modifying the dimensions of the structure while keeping the noise level in the cavity and other structural design criteria within specified limits.

Brama, Torsten↗
NASA NTRS2003

Hierarchical Image Segmentation of Remotely Sensed Data using Massively Parallel GNU-LINUX Software

A hierarchical set of image segmentations is a set of several image segmentations of the same image at different levels of detail in which the segmentations at coarser levels of detail can be produced from simple merges of regions at finer levels of detail. In [1], Tilton, et a1 describes an approach for producing hierarchical segmentations (called HSEG) and gave a progress report on exploiting these hierarchical segmentations for image information mining. The HSEG algorithm is a hybrid of region growing and constrained spectral clustering that produces a hierarchical set of image segmentations based on detected convergence points. In the main, HSEG employs the hierarchical stepwise optimization (HSWO) approach to region growing, which was described as early as 1989 by Beaulieu and Goldberg. The HSWO approach seeks to produce segmentations that are more optimized than those produced by more classic approaches to region growing (e.g. Horowitz and T. Pavlidis, [3]). In addition, HSEG optionally interjects between HSWO region growing iterations, merges between spatially non-adjacent regions (i.e., spectrally based merging or clustering) constrained by a threshold derived from the previous HSWO region growing iteration. While the addition of constrained spectral clustering improves the utility of the segmentation results, especially for larger images, it also significantly increases HSEG s computational requirements. To counteract this, a computationally efficient recursive, divide-and-conquer, implementation of HSEG (RHSEG) was devised, which includes special code to avoid processing artifacts caused by RHSEG s recursive subdivision of the image data. The recursive nature of RHSEG makes for a straightforward parallel implementation. This paper describes the HSEG algorithm, its recursive formulation (referred to as RHSEG), and the implementation of RHSEG using massively parallel GNU-LINUX software. Results with Landsat TM data are included comparing RHSEG with classic region growing.

Tilton, James C.↗
NASA NTRS2008

Rapid Turnaround of Costing/Designing of Space Missions Operations

The Ground Segment Team (GST), at NASA's Jet Propulsion Laboratory in Pasadena, California, provides high-level mission operations concepts and cost estimates for projects that are in the formulation phase. GST has developed a tool to track costs, assumptions, and mission requirements, and to rapidly turnaround estimates for mission operations, ground data systems, and tracking for deep space and near Earth missions. Estimates that would often take several weeks to generate are now generated in minutes through the use of an integrated suite of cost models. The models were developed through interviews with domain experts in areas of Mission Operations, including but not limited to: systems engineering, payload operations, tracking resources, mission planning, navigation, telemetry and command, and ground network infrastructure. Data collected during interviews were converted into parametric cost models and integrated into one tool suite. The tool has been used on a wide range of missions from small Earth orbiters, to flagship missions like Cassini. The tool is an aid to project managers and mission planners as they consider different scenarios during the proposal and early development stages of their missions. The tool is also used for gathering cost related requirements and assumptions and for conducting integrated analysis of multiple missions.

Cost estimation↗
NASA NTRS1980

A semianalytical satellite theory for weak time-dependent perturbations

The modifications of the semianalytical satellite theory required to include these 'weak' time dependent perturbations are described. The new formulation results in additional terms in the short periodic variations but does not change the averaged equations of motion. Thus the m monthly terms are still included in the averaged equations of motion. This contrasts with the usual approach for the strongly time dependent perturbations in which the m monthly (or m daily, if tesseral harmonics are being considered) terms would be eliminated from the averaged equations of motion and included in the short periodics computation. Numerical test results for the GPS case obtained with a numerical averaging implementation of the new theory demonstrate the accuracy improvement.

Cefola, P.↗
NASA NTRS1989

Fuzzy set applications in engineering optimization: Multilevel fuzzy optimization

A formulation for multilevel optimization with fuzzy objective functions is presented. With few exceptions, formulations for fuzzy optimization have dealt with a one-level problem in which the objective is the membership function of a fuzzy set formed by the fuzzy intersection of other sets. In the problem examined here, the goal set G is defined in a more general way, using an aggregation operator H that allows arbitrary combinations of set operations (union, intersection, addition) on the individual sets Gi. This is a straightforward extension of the standard form, but one that makes possible the modeling of interesting evaluation strategies. A second, more important departure from the standard form will be the construction of a multilevel problem analogous to the design decomposition problem in optimization. This arrangement facilitates the simulation of a system design process in which different components of the system are designed by different teams, and different levels of design detail become relevant at different time stages in the process: global design features early, local features later in the process.

Diaz, Alejandro R.↗
NASA NTRS1961

A Feasible Approach for an Early Manned Lunar Landing. Part II: Detailed Report of Ad Hoc Task Group

This report, in two parts, presents a program development plan for attempting a first manned lunar landing in 1967. The two parts consist of a Summary Report and a Detailed Report representing the coordinated output of the Ad Hoc Task Group assigned to the study. The study was started in response to the request for such a study by the Associate Administrator in his memorandum of May 2, 1961 establishing the Ad Hoc Task Group. The purpose of the study was to take a first cut at the tasks associated with the design, development and construction of the equipment and facilities as well as the development of the crews, and to show the time phasing 6f these tasks. Included are the space sciences, life science and advanced technology tasks whose data and results are needed for designing and developing the systems required in carrying out the mission. The plan presented in the two reports does not presume to be a firm plan. Its basic purpose is, by choosing one feasible method, to size up the scope, schedule and cost of the job, discover the main problems, pacing items and major decisions and provide a threshold from which a firm and detailed project development plan can be jointly formulated by the various elements of NASA.

Fleming, William A.↗
DOE OSTI2020

Insights into the Structure and Energy of DNA Nanoassemblies

Since the pioneering work of Ned Seeman in the early 1980s, the use of the DNA molecule as a construction material experienced a rapid growth and led to the establishment of a new field of science, nowadays called structural DNA nanotechnology. Here, the self-recognition properties of DNA are employed to build micrometer-large molecular objects with nanometer-sized features, thus bridging the nano- to the microscopic world in a programmable fashion. Distinct design strategies and experimental procedures have been developed over the years, enabling the realization of extremely sophisticated structures with a level of control that approaches that of natural macromolecular assemblies. Nevertheless, our understanding of the building process, i.e., what defines the route that goes from the initial mixture of DNA strands to the final intertwined superstructure, is, in some cases, still limited. In this review, we describe the main structural and energetic features of DNA nanoconstructs, from the simple Holliday junction to more complicated DNA architectures, and present the theoretical frameworks that have been formulated until now to explain their self-assembly. Deeper insights into the underlying principles of DNA self-assembly may certainly help us to overcome current experimental challenges and foster the development of original strategies inspired to dissipative and evolutive assembly processes occurring in nature.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2023

The Design, Verification and Performance of the James Webb Space Telescope

The James Webb Space Telescope (JWST) is NASA’s flagship mission successor to the highly successful Hubble Space Telescope. It is an infrared observatory featuring a cryogenic 6.6 m aperture, deployable Optical Telescope Element (OTE) with a payload of four science instruments (SIs) assembled into an Integrated Science Instrument Module (ISIM) that provide imagery and spectroscopy in the near-infrared band between 0.6 and 5 μm and in the mid-infrared band between 5 and 28.1 μm. JWST was successfully launched on 2021 December 25 aboard an Ariane 5 launch vehicle. All 50 major deployments were successfully completed on 2022 January 8. The observatory performed all midcourse correction maneuvers and achieved its operational mission orbit around the Sun–Earth second Lagrange point (L2). All commissioning and calibration activities have been completed, and JWST has begun its science mission. This paper will provide a description of the driving requirements and their technical challenges, the engineering processes involved in the design formulation, the resulting observatory design, the verification programs that proved it to be flightworthy, and the measured on-orbit performance of the observatory. Since companion papers will describe the details of the OTE and SIs, this paper will concentrate on describing the key features of the observatory architecture that accommodates these elements, particularly those features and capabilities associated with accommodating the radiometric and image-quality performance.

Galaxy evolution↗
DOE OSTI2023

Variations in Creep Performance of P92 Steel Within Its Composition Range: Influence of N Solubility

Underspecified composition ranges often lead to alloys with unpredictable mechanical performance. To better understand the changes in microstructure and mechanical performance associated with variations of key elements, three versions of P92 are formulated within, or close to, the specified allowable for N, B, and C ranges. Chromium and Si are also varied to influence N solubility. Different service conditions (i.e., temperature and stress) are explored. It is observed that >80% decrease in creep life occurs at 625 °C and 155 MPa for the highest B and N containing alloy. Multiscale characterization reveals key changes due to the trace element variation. The high B and N containing alloy forms deleterious BN precipitates with morphology that promotes crack nucleation and damage accumulation, but this alloy additionally forms higher fractions of beneficial MX precipitates. The alloy with the lowest B and N concentrations but greater C content shows the best creep performance—a consequence of the refined M 23 C 6 carbide precipitate population and the absence of large-scale inclusions or BN precipitates. Calculations of creep activation energy reveal that the high B and N containing alloy is more prone to damage accumulation which causes an early onset of accelerating creep and greater minimum creep rate.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Preheating in Palatini Higgs inflation on the lattice

We study preheating following Higgs inflation in the Palatini formulation of gravity. We numerically evolve perturbations of the radial mode of the Higgs field and that of three scalars modeling the gauge bosons. We compare the two non-perturbative mechanisms of growth of excitations — parametric resonance and tachyonic instability — and confirm that the latter plays the dominant role. Our results provide further evidence that preheating in Palatini Higgs inflation happens within a single oscillation of the Higgs field about the bottom of its potential, consistent with the approximation of an instantaneous preheating.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1993

Development and use of computational techniques in Army Aviation research and development programs for crash resistant helicopter technology

During the 1960's over 30 full-scale aircraft crash tests were conducted by the Flight Safety Foundation under contract to the Aviation Applied Technology Directorate (AATD) of the U.S. Army Aviation Systems Command (AVSCOM). The purpose of these tests were to conduct crash injury investigations that would provide a basis for the formulation of sound crash resistance design criteria for light fixed-wing and rotary wing aircraft. This resulted in the Crash Survival Design Criteria Designer's Guide which was first published in 1967 and has been revised numerous times, the last being in 1989. Full-scale aircraft crash testing is an expensive way to investigate structural deformations of occupied spaces and to determine the decelerative loadings experienced by occupants in a crash. This gave initial impetus to the U.S. Army to develop analytical methods to predict the dynamic response of aircraft structures in a crash. It was believed that such analytical tools could be very useful in the preliminary design stage of a new helicopter system which is required to demonstrate a level of crash resistance and had to be more cost effective than full-scale crash tests or numerous component design support tests. From an economic point of view, it is more efficient to optimize for the incorporation of crash resistance features early in the design stage. However, during preliminary design it is doubtful if sufficient design details, which influence the exact plastic deformation shape of structural elements, will be available. The availability of simple procedures to predict energy absorption and load-deformation characteristics will allow the designer to initiate valuable cost, weight, and geometry tradeoff studies. The development of these procedures will require some testing of typical specimens. This testing should, as a minimum, verify the validity of proposed procedures for providing pertinent nonlinear load-deformation data. It was hoped that through the use of these analytical models, the designer could optimize aircraft design for crash resistance from both a weight and cost increment standpoint, thus enhancing the acceptance of the design criteria for crash resistance.

Burrows, Leroy T.↗
DOE OSTI2022

Quantization: History and problems

In this work, I explore the concept of quantization as a mapping from classical phase space functions to quantum operators. I discuss the early history of this notion of quantization with emphasis on the works of Schrödinger and Dirac, and how quantization fit into their overall understanding of quantum theory in the 1920's. Dirac, in particular, proposed a quantization map which should satisfy certain properties, including the property that quantum commutators should be related to classical Poisson brackets in a particular way. However, in 1946, Groenewold proved that Dirac's mapping was inconsistent, making the problem of defining a rigorous quantization map more elusive than originally expected. This result, known as the Groenewold-Van Hove theorem, is not often discussed in physics texts, but here I will give an account of the theorem and what it means for potential ``corrections” to Dirac's scheme. Other proposals for quantization have arisen over the years, the first major one being that of Weyl in 1927, which was later developed by many, including Groenewold, and which has since become known as Weyl Quantization in the mathematical literature. Another, known as Geometric Quantization, formulates quantization in differential-geometric terms by appealing to the character of classical phase spaces as symplectic manifolds; this approach began with the work of Souriau, Kostant, and Kirillov in the 1960's. I will describe these proposals for quantization and comment on their relation to Dirac's original program. Along the way, the problem of operator ordering and of quantizing in curvilinear coordinates will be described, since these are natural questions that immediately present themselves when thinking about quantization.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗