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Physics Flash: Summer 2024

Each year, the Lab’s Partnerships and Pipeline Office's Postdoc Program honors outstanding efforts made by postdocs and mentors that have led to a positive impact on the Lab and its missions. For such contributions, Applied and Fundamental Physics (P-2) researchers David Rivera and Sowjanya Gollapinni have been recognized by the program.

99 GENERAL AND MISCELLANEOUS↗

Convergence Criteria for Multiphysics Simulations

The behavior of engineered systems is often influenced by multiple physical phenomena, such as mechanical deformation, heat transfer, and chemical species transport and reactions. There are often strong interactions between these phenomena, and there is increasing interest in applying coupled-physics models to improve understanding of physical behavior under complex environmental conditions. Multiple simulation frameworks that facilitate coupled-physics simulations are in widespread use, and these employ a variety of techniques to account for interactions between those physics. Many frameworks solve the physics models independently and transfer results between them. Alternatively, a single monolithic system of equations for every physics model can be formed and solved. Each of these approaches has its benefits and drawbacks, and the optimal approach varies depending on the nature of the problem. The open-source MOOSE framework was developed targeting solution of large-scale multiphysics problems. Although it provides options for all these coupling approaches, its standard approach for multiphysics solutions is to form and solve a single monolithic system of equations containing the unknowns for all physics models. MOOSE provides a streamlined approach for users to define the solution variables, the terms in the partial differential equations pertaining to each variable, and interactions between solution variables. One aspect of the monolithic solution approach that can be problematic, however, is defining appropriate convergence criteria for the nonlinear system. A standard approach is to determine convergence is to simply take a norm of the residual vector corresponding to the full vector of unknowns. However, if the residual vector contains variables for multiple physics models, the magnitudes of those variables can differ significantly, and the variables can converge at significantly different rates from each other. It is important to ensure that the variables for each of the physics are converged, and also ensure that the convergence criteria are not excessively stringent in cases when there is little change in the solution. This talk presents representative multiphysics problems to highlight these issues, and shows strategies for convergence criteria in MOOSE that are robust for multiphysics models under a variety of conditions.

97 - MATHEMATICS AND COMPUTING↗

The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas

JOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic (MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

End-to-End Physics Event Classification with CMS Open Data: Applying Image-Based Deep Learning to Detector Data for the Direct Classification of Collision Events at the LHC

This paper describes the construction of novel end-to-end image-based classifiers that directly leverage low-level simulated detector data to discriminate signal and background processes in pp collision events at the Large Hadron Collider at CERN. To better understand what end-to-end classifiers are capable of learning from the data and to address a number of associated challenges, we distinguish the decay of the standard model Higgs boson into two photons from its leading background sources using high-fidelity simulated CMS Open Data. We demonstrate the ability of end-to-end classifiers to learn from the angular distribution of the photons recorded as electromagnetic showers, their intrinsic shapes, and the energy of their constituent hits, even when the underlying particles are not fully resolved, delivering a clear advantage in such cases over purely kinematics-based classifiers.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Nuclear Power Plant Physical Protection Recommendation Document

This document is aimed at providing guidance to the National Nuclear Security Administration’s (NNSA) Office of International Nuclear Security’s (INS) country and regional teams for implementing effective physical protection systems (PPSs) for nuclear power plants (NPPs) to prevent the radiological consequences of sabotage. This recommendation document includes input from the Physical Protection Functional Team (PPFT), the Response Functional Team (RFT), and the Sabotage Functional Team (SFT) under INS. Specifically, this document provides insights into increasing and sustaining physical protection capabilities at INS partner countries’ NPP sites. Nuclear power plants should consider that the intent of this document is to provide a historical context as well as technologies and methodologies that may be applied to improve physical protection capabilities. It also refers to relevant guidance from the International Atomic Energy Agency (IAEA) and the U.S. Nuclear Regulatory Commission (NRC).

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

L-H transition trigger physics in ITER-similar plasmas with applied n = 3 magnetic perturbations

The L–H transition power threshold PLH is observed to increase with applied n = 3 resonant magnetic perturbations (RMP) in ITER-similar-shape plasmas with balanced neutral beam torque injection in DIII-D. The increase is most pronounced with added electron–cyclotron heating: PLH increases with decreasing edge plasma collisionality as PLH/PLH-08 ~ (ν*)−0.5, where PLH-08 is the 2008 ITPA multi-machine power threshold scaling. This result raises concerns for H-mode access at low edge collisionality in ITER, where RMP may have to be applied before the L–H transition to safely suppress the first edge-localized mode. Non-axisymmetric modifications with RMP include a simultaneous reduction of the radial electric field (Er) well depth and E × B shear. This can be attributed to increasing edge toroidal co-current rotation, and is consistent with substantially increased local long-wavelength turbulence (measured via beam emission spectroscopy). At high RMP perturbation strength the edge electric field Er reverses sign locally (becomes positive), with changes in dominant turbulence modes. Edge magnetic stochasticity provides an attractive explanation of the observed modifications, and the observed changes in toroidal rotation and Er are consistent with a simple fluid model describing radial electron current flow along stochastic fieldlines. The observed collisionality dependence of the L-mode edge electric field with applied RMP is also qualitatively consistent with this model. Reflectometry data indicate a significant reduction of the normalized L-mode radial density gradient a/Ln at high RMP field with simultaneous increase in radial particle flux and electron thermal flux from power balance analysis. We conjecture that the increase of PLH with RMP results from the combined effects of reduced E × B flow shear (increasing turbulent transport levels) and toroidal/poloidal flow modulation due to edge stochasticity. Initial experiments indicate that non-resonant n = 3 magnetic perturbations lead only to relatively small changes in Er, E × B shear and fluctuation characteristics, and have less impact on the L–H transition power threshold. This motivates further exploration of the RMP spectrum dependence of PLH for possible mitigation of the observed threshold increase.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Aerosol Physics for the Lunar Environment: Equations for Lunar Dust Control and Mitigation Technologies

Sticky and jagged dust was ubiquitous during the Apollo missions, causing soiling and abrasion problems with seals, coatings and equipment, in addition to eye irritation and breathing discomfort in the cabin. The Artemis Program of NASA aims to place astronauts on the lunar surface by 2024 and establish a sustainable presence in the following decade. Returning to the Moon requires controlling and mitigating the dust which will be inevitably brought inside the cabins. The state-of-the-science for effective collection of aerosols is based on dynamics of airborne particulate matter under terrestrial conditions. However, the governing physics does not apply to extra-vehicular activity in the hard-vacuum lunar condition. For example, the substantial difference in gravity will dictate particle transport both outside and inside the cabin. In this study, we revisited the aerosol physical phenomena that are assumed in the design of Earth-based aerosol instruments and extend the applicability to different scenarios in lunar missions. As shown, long-term lunar habitats, transfer vehicles to lunar orbital platforms, and low pressure cabin atmospheres have different aerosol dynamics. In all cases, the impact of dust control strategies using gravitational, electrical, and thermal techniques for various mitigation and monitoring hardware is explored. The guidelines provided through this study will show how terrestrial aerosol equipment can translate to lunar dust applications.

Nima Afshar-Mohajer↗

Initial Testing of an In Situ Load Retention Aging Vessel

A thermal aging vessel instrumented with load cells was fabricated. The primary function of the vessel is to continuously monitor the in situ load retention of up to three compressed polymer coupons undergoing thermally accelerated aging under nitrogen. A secondary function is to enable gas sampling of the vessel headspace during thermal aging. Heating of the vessel is achieved using a custom heater jacket. To improve upon our conventional aging study methods which require periodic interruption of aging to perform load testing in an Instron machine at room temperature, this technology aims to automate/facilitate data acquisition/analysis, improve data quality, and enable uninterrupted compression of the polymer which represents the service condition. As an example case to assess functionality of the in situ vessel, the load retention of a siloxane elastomer material additively manufactured by direct-ink-writing (DIW) was measured at three different isothermal aging temperatures for ~1 month. Initial compression of the coupons while near the aging temperature was achieved by temporarily opening the heated vessel to access the interior chamber and manually tightening four nuts to drive the heated compression plate down onto the heated coupons. Initial testing demonstrated achievement of the primary load retention monitoring function. Unfortunately, the vessel leaked which prevented gas sampling; an active purge was used to maintain a nitrogen atmosphere. Welded or otherwise sealed joints, which could be implemented in a future design, would likely eliminate leak paths. To apply time-temperature superposition (TTS), a technique used to provide long-term prediction of the load retention from short-term isothermal data, the load retention needed to be calculated relative to the load at an estimated “equilibrium” time, after most of the transient viscoelastic physical relaxation occurred. The peak load immediately after compression could not be used as the load retention basis for two reasons: (1) age-related changes must be isolated from non-age-related physical relaxation before applying TTS and (2) the manual mechanism used to compress the specimens at the aging temperature was neither smooth nor repeatable which affected the peak load value. To better understand the effect of the mode of initial compression on the measured load, and possibly better estimate “equilibrium” physical relaxation times, systematic stress relaxation experiments were performed using an Instron machine with a thermal chamber. At a given temperature, the DIW polymer was compressed to a fixed strain in either a stepped or continuous manner at two different rates, then held at that strain for 24 hrs. The results indicated that, at a given temperature, the different stress relaxation curves appeared to converge to the same curve at some “equilibrium” time when the non-age-related physical relaxation was mostly complete. Though this observation suggests that the discontinuous manual compression employed by the vessel is feasible, a compression mechanism that is rapid, smooth, and repeatable would enhance its use.

36 MATERIALS SCIENCE↗

Gearbox bearing crack growth prognostics and uncertainty quantification with physics-informed machine learning

This paper introduces the extreme theory of functional connections (X-TFC), a physics-informed machine learning algorithm, and tailors it to estimate the remaining useful life (RUL) of wind turbine gearbox bearings experiencing fatigue crack growth. Unlike purely data-driven methods, X-TFC embeds a physics model, based on Head's theory in this work, into its training objective. The core of X-TFC is a random-projection single-layer neural network trained via an extreme learning machine, which requires only limited damage progression data and solves for output weights with a least-squares optimization algorithm. A composite loss function balances the network's fit to observed degradation data against the residuals of the governing crack growth differential equation, ensuring the learned damage trajectory remains physically plausible. When applied to a vibration-based health-index (HI) dataset measured during the growth of a crack on the inner ring of a high-speed bearing in a wind turbine gearbox (Bechhoefer and Dubé, 2020), X-TFC achieves near-zero prediction bias. Even when trained on only the first 10 %–20 % of the damage progression data, with sufficient physics weighting its predictions remain monotonic and smooth, delivering high prognosability and trendability. To quantify the epistemic uncertainty, we employ a Monte Carlo ensemble of independently initialized X-TFC models trained on noise-perturbed data, which yields confidence intervals around each RUL estimate and captures both model-parameter and epistemic uncertainty. In addition to a vibration-based HI, we demonstrate that the proposed framework can be directly applied to a supervisory control and data acquisition (SCADA) data-based HI (Eftekhari Milani et al., 2026) measured during similar wind turbine gearbox bearing crack faults, preserving its accuracy and interpretability. This extension shows the versatility of our approach, which is applicable to bearings of multiple gearbox manufacturers, models, and ratings using only SCADA data. By integrating domain knowledge with machine learning, X-TFC offers a rapid, reliable tool for crack prognostics. Its adaptability to other bearing failure modes, such as pitch bearing ring cracks, positions X-TFC as a powerful enabler of data-driven, physics-informed asset management in the wind energy sector and beyond.

17 WIND ENERGY↗

Physical interpretation and development of ultrasonic nondestructive evaluation techniques applied to the quantitative characterization of textile composite materials

The development and implementation of advanced ultrasonic nondestructive evaluation methods applied to the characterization of composite materials requires a better understanding of the physics underlying the interaction of ultrasound with the material. The purpose of this investigation is to identify and characterize the features of complex, three dimensional materials that limit the ability of ultrasound to detect flaws in this broad class of emerging materials. In order to explore the interaction of ultrasound with such complex media, we investigate the characteristics of ultrasonic fields which have propagated through samples with complex geometries and/or internal architecture. We focus on the physics that underlies the detection of flaws in such materials.

Miller, James G.↗

Informing nuclear physics via machine learning methods with differential and integral experiments

Information from differential nuclear-physics experiments and theory is often too uncertain to accurately define nuclear-physics observables such as cross sections or energy spectra. Integral experimental data, representing the applications of these observables, are often more precise but depend simultaneously on too many of them to unambiguously identify issues in the observable with human expert analysis alone. Here, we explore how we can leverage physics knowledge gained from differential experimental data, nuclear theory, integral experiments, and neutron-transport calculations to better understand nuclear-physics observables in the context of the application area represented by integral experiments. We support this task with machine-learning methods to discern trends in a large amount of convoluted data. Differential and integral information was used in an analysis augmented by the random forest and the Shapley additive explanations metric. We chose as an application area one that is represented by criticality measurements and pulsed-sphere neutron-leakage spectra. We show one representative example ( 241 Pu fission observables) where the combination of differential and integral information allowed to resolve issues in data representing these observables. As a starting point, the machine learning (ML) algorithms highlighted several observables as leading potentially to bias in simulating integral experiments. Differential information, paired with sensitivity to integral quantities, allowed us then to pinpoint one specific observable ( 241 Pu fission cross section) as the main driver of bias. The comparison to integral experiments, on the other hand, allowed us to indicate a likely reliable experiment among several discrepant ones for this observables. In other cases (e.g., 239 Pu observables), we were not able to resolve the confounding introduced by integral experiments but instead highlighted the need for targeted new experiments and theory developments to better constrain the nuclear-physics space for the application area represented by integral experiments. We were able to combine information from differential experimental data, nuclear-physics theory, integral experiments, and neutron-transport simulations of the latter experiments with the help of the random forest algorithm and expert judgment. This combination of knowledge allows to improve our description of nuclear-physics observables as applied to a particular application area.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Process-oriented stochastic perturbations applied to the parametrization of turbulence and shallow convection for ensemble prediction

Weather forecasting nowadays often requires some estimation of uncertainties associated with the output of meteorological models, in order to better inform decision making, especially in the context of intense weather events. Ensemble prediction systems provide such information through sets of possible scenarios which are designed to represent various uncertainty sources, including model uncertainties. A wide variety of methods have been proposed to estimate model uncertainties, among which perturbation methods targeting uncertain processes are a promising research field. Here, we focus on the representation of small-scale variability by process-oriented perturbation schemes applied to two key physical processes, namely turbulence and shallow convection. The perturbations are applied to a single-column version of the convection-permitting AROME model, in three idealized boundary-layer cases. Large-eddy simulations (LESs) of the same cases serve as a reference for the subgrid variability that has to be represented, and the results are also compared to those given by the Stochastically Perturbed Parametrization Tendencies (SPPT) method, which is a method commonly used by weather forecast centres to represent model uncertainty. The spread produced by our process-oriented perturbations of turbulence and shallow convection does not represent all the small-scale variability implied by the LESs for temperature and humidity. However, it is of a similar order of magnitude for the wind, thanks to perturbations generated by the stochastic turbulence scheme. The dispersion is structurally different from what is obtained with SPPT. It is non-negligible in the lower levels, where SPPT perturbations are usually suppressed because of numerical instabilities, indicating a possible complementarity between the schemes.

54 ENVIRONMENTAL SCIENCES↗

On the Recoverability of Reactor Dynamics from Point Kinetics Data using SINDYc

Data driven models for reactor dynamics tend to face a few notable challenges. First, the broad range of timescales present across the various feedback mechanisms. Next, high standards for safety and importance of model performance in all operational domains. Finally, the correlations between state variables observed in most transients leads to difficulties when methods attempt to attribute certain dynamic phenomena to a particular cause. The current paper seeks to support efforts towards incorporating physics knowledge into one particular data-driven method for finding reactor dynamics called ``Sparse Identification of Nonlinear Dynamics with Control (SINDYc). The incorporation of physical knowledge into SINDYc will help address the challenges listed above by giving the model an initial understanding of the system being modeled. The demonstration of an exact representation of a set of point reactor dynamics equations in SINDYc is provided. Then, this allows for further discussion with mathematical justification as to why SINDYc, and other data-driven methods, may be difficult to apply to nuclear reactor dynamics due to correlations in the state variables. A particularly useful result of this work is the set of candidate functions required in SINDYc to exactly represent the reactor dynamics under a point approximation. In the future, SINDYc can be integrated with point models for reactor dynamics before being applied to the physical system to yield higher accuracy.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

On the Effects of Surface Roughness on Boundary Layer Transition

Surface roughness can influence laminar-turbulent transition in many different ways. This paper outlines selected analyses performed at the NASA Langley Research Center, ranging in speed from subsonic to hypersonic Mach numbers and highlighting the beneficial as well as adverse roles of the surface roughness in technological applications. The first theme pertains to boundary-layer tripping on the forebody of a hypersonic airbreathing configuration via a spanwise periodic array of trip elements, with the goal of understanding the physical mechanisms underlying roughness-induced transition in a high-speed boundary layer. The effect of an isolated, finite amplitude roughness element on a supersonic boundary layer is considered next. The other set of flow configurations examined herein corresponds to roughness based laminar flow control in subsonic and supersonic swept wing boundary layers. A common theme to all of the above configurations is the need to apply higher fidelity, physics based techniques to develop reliable predictions of roughness effects on laminar-turbulent transition.

Choudhari, Meelan M.↗

Breakdown front dynamics of streamer-like discharge in electron-charged polymethyl methacrylate

We present the results of an experimental study of fast breakdown in electron-charged polymethyl methacrylate. We irradiate bulk polymethyl methacrylate disks with diameters up to one meter at different implanted charge densities and measure the discharge current during the forced electrical breakdown of the material. We infer the breakdown dynamics from these current waveforms, including the velocity and time dependence of the electric field driving the breakdown, and compare these results with the physical electrical tree patterns left behind in the material. We find that the dynamics and physical characteristics of the breakdown channels in electron-irradiated solids depart from typical expectations of electrical treeing behavior in solid materials. Thus, we interpret these results as an expression of streamer discharges in dense gases motivated by the existence of trapped gases in the solid due to radiation damage. We show that the dynamics of the breakdown channels in the electron-charged solid dielectric material is well described by applying standard streamer physics to this physical system. The results show that the dynamics of breakdown channels in solid dielectric material is a promising avenue to further understand streamer discharges in different media under extreme conditions.

36 MATERIALS SCIENCE↗

Foam on Tile Impact Modeling for the Space Shuttle Program

Following the breakup of the Space Shuttle Columbia during reentry a NASA-wide investigation team was formed to examine the probable damage inflicted on Orbiter Thermal Protection System (TPS) elements by impact of External Tank insulating foam projectiles. Our team was to apply rigorous, physics-based analysis techniques to help determine parameters of interest for an experimental test program, utilize validated codes to investigate the full range of impact scenarios, and use analysis derived models to predict aero-thermal-structural responses to entry conditions. We were to operate on a non-interference basis with the j Team, and were to supply significant findings to that team and to the Orbiter Vehicle Engineering Working Group, being responsive to any solicitations for support from these entities. The authors formed a working sub-group within the larger team to apply the Smooth Particle Hydrodynamics code SPHC to the damage estimation problem. Numerical models of the LI-900 TPS tiles and of the BX-250 foam were constructed and used as inputs into the code. Material properties needed to properly model the tiles and foam were obtained from other working sub-groups who performed tests on these items for this purpose. Two- and three- dimensional models of the tiles were constructed, including the glass outer layer, the densified lower layer of LI-900 insulation, the Nomex felt Strain Isolation Pad (SIP) mounting layer, and the underlying aluminum 2024 vehicle skin. A model for the BX-250 foam including porous compression, elastic rebound, and surface erosion was developed. Code results for the tile damage and foam behavior were extensively validated through comparison with the Southwest Research Institute (SwRI) foam-on-tile impact experiments carried out in 1999. These tests involved small projectiles striking individual tiles and small tile arrays. Following code and model validation we simulated impacts of larger ET foam projectiles on the TPS tile systems used on the wings of the orbiter. Tiles used on the Wing Acreage, the Main Landing Gear Door, and the Carrier Panels near the front edge of the wing were modeled. Foam impacts shot for the CAB investigation were modeled, as well as impacts at larger angles, including rapid rotation of the projectile, and with varying foam properties. General results suggest that foam impacts on tiles at about 500 mph could cause appreciable damage if the impact angle is greater than about 20 degrees. Some variations of the foam properties, such as increased brittleness or increased density could increase damage in some cases. Rapid (17 rps) rotation failed to increase the damage for the two cases considered. This does not rule out other cases in which the rotational energy might lead to an increase in tile damage, but suggests that in most cases rotation will not be an important factor. Similar models will be applied for other impacting materials, other velocities, and other geometries as part of the Return to Flight process.

Stellingwerf, R. F.↗

Towards Predictive Plasma Science and Engineering through Revolutionary Multi-Scale Algorithms and Models (Final Report)

This report describes the high-level accomplishments from the Plasma Science and Engineering Grand Challenge LDRD at Sandia National Laboratories. The Laboratory has a need to demonstrate predictive capabilities to model plasma phenomena in order to rapidly accelerate engineering development in several mission areas. The purpose of this Grand Challenge LDRD was to advance the fundamental models, methods, and algorithms along with supporting electrode science foundation to enable a revolutionary shift towards predictive plasma engineering design principles. This project integrated the SNL knowledge base in computer science, plasma physics, materials science, applied mathematics, and relevant application engineering to establish new cross-laboratory collaborations on these topics. As an initial exemplar, this project focused efforts on improving multi-scale modeling capabilities that are utilized to predict the electrical power delivery on large-scale pulsed power accelerators. Specifically, this LDRD was structured into three primary research thrusts that, when integrated, enable complex simulations of these devices: (1) the exploration of multi-scale models describing the desorption of contaminants from pulsed power electrodes, (2) the development of improved algorithms and code technologies to treat the multi-physics phenomena required to predict device performance, and (3) the creation of a rigorous verification and validation infrastructure to evaluate the codes and models across a range of challenge problems. These components were integrated into initial demonstrations of the largest simulations of multi-level vacuum power flow completed to-date, executed on the leading HPC computing machines available in the NNSA complex today. These preliminary studies indicate relevant pulsed power engineering design simulations can now be completed in (of order) several days, a significant improvement over pre-LDRD levels of performance.

42 ENGINEERING↗

Ergodic Lagrangian dynamics in a superhero universe

We present a fictional scenario that, while undeniably whimsical, provides the foundation for a unique exercise in extended problem solving, physics analysis, and quantitative model development. Starting with the foundational premise of the Wild Cards shared-world superhero universe, we demonstrate how a variety of concepts appropriate to the advanced undergraduate level—ergodicity, functional analysis, Lagrangian mechanics, and the ever-important simplifying approximation—can be combined into a rich, coherent mathematical model. The goal of this case study is to develop a useful pedagogical exercise in exploring an open-ended research question that presents, at first glance, no clear path forward. Being both eclectic and lengthy, this exercise offers a unique way for students to apply their core physics and mathematics education. It is perhaps best used within a senior honors seminar or within a brief (e.g., January term) elective class.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗