THE MOTION OF SATELLITE 1958 EPSILON AROUND ITS CENTER OF MASS
Motion of satellite 1958 epsilon around its center of mass
EXPLORER IV SATELLITE↗
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Pursuit Eye Movements
When viewing objects, primates use a combination of saccadic and pursuit eye movements to stabilize the retinal image of the object of regard within the high-acuity region near the fovea. Although these movements involve widespread regions of the nervous system, they mix seamlessly in normal behavior. Saccades are discrete movements that quickly direct the eyes toward a visual target, thereby translating the image of the target from an eccentric retinal location to the fovea. In contrast, pursuit is a continuous movement that slowly rotates the eyes to compensate for the motion of the visual target, minimizing the blur that can compromise visual acuity. While other mammalian species can generate smooth optokinetic eye movements - which track the motion of the entire visual surround - only primates can smoothly pursue a single small element within a complex visual scene, regardless of the motion elsewhere on the retina. This ability likely reflects the greater ability of primates to segment the visual scene, to identify individual visual objects, and to select a target of interest.
Krauzlis, Rich↗
Evaluation of Intranasal Scopolamine for the Prevention of Wave Motion Induced Motion Sickness While Maintaining Performance on Operationally Relevant Tasks
Motion sickness represents one of the greatest clinical challenges impacting crew activities during and following g-transitions. Our overall goal is to characterize the effectiveness of motion sickness countermeasures during controlled laboratory experiments using capsule wave motion simulation and during field testing in operational environments. This laboratory study focused on prevention of motion sickness using intranasal scopolamine using a double-blinded repeated measures design in 30 subjects (19M, 11F). Intranasal scopolamine was provided by Defender Pharmaceuticals, Inc. (DPI-386 Nasal Gel, referred to as Inscop) self-administered by a nasal pump (Aptar Pharma) that delivers 0.4 mg dose (0.2 mg / nostril). During each session, subjects were exposed to complex wave motion on a six degree-of-freedom platform that included pitch, roll and heave at provocative stimulus frequencies (0.1-0.25 Hz) while seated in an illuminated cabin deprived of external visual cues. Motion sickness symptoms were compared across treatment and placebo control sessions counterbalanced across subjects and separated by at least one week. The time-to-motion sickness endpoint was based on severe malaise, defined as symptom score of ≥8 points using the Pensacola Diagnostic Index [1]. The bioavailability of scopolamine for each session was estimated from plasma concentrations obtained every 15 min[2]. Side effects during the treatment session were minimal, and performance was not impaired on a test battery including motion perception tracking, tablet-based eye-hand coordination and psychomotor vigilance testing. The plasma concentration remained near-peak levels throughout the 45 min motion sickness testing for most subjects. The percentile ranking on the Motion Sickness Susceptibility Questionnaire [MSSQ, 3]was moderately correlated with the motion sickness time-to-endpoint during the placebo control session (rho = –0.3, p=0.056). Seventeen subjects did not reach an endpoint during their placebo session and were eliminated from subsequent analysis. Another subject was excluded due to insufficient plasma concentration during the treatment session. For the remaining 12 subjects, the change in time-to-motion sickness endpoint between placebo and treatment sessions was moderately correlated with plasma concentration (rho =0.48, p = 0.056), improving on average 4.4 ± 18.4 min, mean ± std with Inscop versus placebo. Our results are consistent with previous findings that intranasal delivery of scopolamine can be effective at reducing motion sickness symptoms with minimal cognitive or sedative side effects. Future work is needed to optimize the delivery of Inscop for rescue (treatment) of symptoms following g-transitions as one of the key advantages of this formulation is self-administration in a suited environment. We will also explore how the combination of Inscop with non-pharmaceutical sensory aids, e.g., vibrotactile feedback of Earth vertical, may further mitigate motion sickness and improve task performance.
S J Wood↗
Electron collection by a highly positive satellite in the ionosphere: Test aprticle simulation
Collection of electrons by a satellite at a relatively high potential in low Earth orbit, including the effects of the satellite's orbital motion, remains a poorly understood phenomenon. Using a test particle simulation in which charged particle motion is tracked in prescribed electric fields, we calculate here the current collection and the nature of the energy distribution function of the electrons collected by the satellite, including the effects of the satellite orbital motion. Calculations of the collected current without the orbital motion show an excellent agreement with the current from the Parker-Murphy model, but with the orbital motion of the satellite the current collection is enhanced, the degree of enhancement depending on the size of the satellite sheath extending along the magnetic field line. In the latter case, the flow of electrons around the satellite shows some interesting behaviors including azimuthal E x B drift around the satellite, axial trapping along the magnetic field, and formation of field-aligned flow of electrons in the wake region. The total energy of the collected electrons is approximately equal e phi(sub 0), where phi(sub 0) is the satellite potential, but the partition of the energy into components parallel (W(sub parallel)) and perpendicular (W(sub perpendicular)) to the magnetic field shows interesting features. For the magnetic field along the polar axis Z, the energy distributions near the poles are perfectly field aligned, that is, the parallel energy (W(sub parallel) = e phi(sub 0). The perpendicular component (W(sub perpendicular)) progressively increases toward the 'equator' of the spherical satellite. For a sufficiently large bias potential phi(sub 0), the equatorial energy distributions f(W(sub parallel)) and f(W(sub perpendicular)) show fine structures having multiple peaks. These features of the distribution functions result from the chaotic motion of electrons and the associated coupling between the electron motions parallel and perpendicular to the magnetic field in strongly nonuniform electric fields near the satellite.
Singh, Nagendra↗
General Equations of Motion for a Damaged Asymmetric Aircraft
There is a renewed interest in dynamic characteristics of damaged aircraft both in order to assess survivability and to develop control laws to enhance survivability. This paper presents a set of flight dynamics equations of motion for a rigid body not necessarily referenced to the body's center of mass. Such equations can be used when the body loses a portion of its mass and it is desired to track the motion of the body s previous center of mass/reference frame now that the mass center has moved to a new position. Furthermore, results for equations presented in this paper and equations in standard aircraft simulations are compared for a scenario involving a generic transport aircraft configuration subject to wing damage.
Bacon, Barton J.↗
Digital Image Correlation for Performance Monitoring
Evaluating the health of a mechanism requires more than just a binary evaluation of whether an operation was completed. It requires analyzing more comprehensive, full-field data. Health monitoring is a process of non-destructively identifying characteristics that indicate the fitness of an engineered component. In order to monitor unit health in a production setting, an automated test system must be created to capture the motion of mechanism parts in a real-time and non-intrusive manner. One way to accomplish this is by using high-speed video and Digital Image Correlation (DIC). In this approach, individual frames of the video are analyzed to track the motion of mechanism components. The derived performance metrics allow for state-of-health monitoring and improved fidelity of mechanism modeling. The results are in-situ state-of-health identification and performance prediction. This paper introduces basic concepts of this test method, and discusses two main themes: the use of laser marking to add fiducial patterns to mechanism components, and new software developed to track objects with complex shapes, even as they move behind obstructions. Finally, the implementation of these tests into an automated tester is discussed.
Palaviccini, Miguel↗
Version 2 - Tidal Energy Resource Characterization, Velocity and Turbulence Measurements, Cook Inlet, AK, 2021
Version 2 of MHKDR submission #475, which also contains the metadata for these measurements. Files here are stored in netCDF4 format, which can be read in Python using the "netcdf4" or "xarray" packages or in MATLAB using the "ncread()" function. Changes are as follows: MWM1 (mid-water mooring 1) ADV - Time length of data now ends when bottom track data is lost - Improved filtering and quality control of bottom track measurements - improved motion correction of velocity - Improved Doppler noise subtraction - improved velocity, turbulent kinetic energy (TKE), Reynolds stress, time interval (TI), and dissipation rate estimations MWM1 Down-Looking ADCP - Improved filtering and quality control of bottom track measurements - improved motion correction of velocity - Improved Doppler noise subtraction - improved TI and dissipation rate estimations - Removed TKE calculations - accuracy could not be verified MWM1 Up-Looking ADCP - Improved Doppler noise subtraction - improved TI and dissipation rate estimations - Removed TKE calculations - accuracy could not be verified - Velocity unchanged MWM2 ADV - Removes data after instrument was hit by debris - Improved filtering and quality control of bottom track measurements - improved motion correction of velocity - Improved Doppler noise subtraction - improved velocity, TKE, Reynolds stress, TI, and dissipation rate estimations MWM2 Down-Looking ADCP - Improved filtering and quality control of bottom track measurements - improved motion correction of velocity - Improved Doppler noise subtraction - improved TI and dissipation rate estimations - Removed TKE calculations - accuracy could not be verified MWM2 Up-Looking ADCP - Improved filtering and quality control of bottom track measurements - improved motion correction of velocity - Improved Doppler noise subtraction - improved TI and dissipation rate estimations - Removed TKE calculations - accuracy could not be verified THEOM (bottom lander) ADV - Improved Doppler noise subtraction - improved TKE, Reynolds stress, TI, and dissipation rate estimations - Velocity unchanged THEOM ADCP (up-looking) - Improved Doppler noise subtraction - improved TKE, TI, and dissipation rate estimations - Removed vertical TKE component (w'w' bar) - Velocity unchanged
16 TIDAL AND WAVE POWER↗
An Intelligent Vision-only Operator Interface for Dexterous Robots
This paper describes the development of a vision-only intelligent interface for dexterous robots capable of tracking operator motions, analyzing trajectory feasibility, and providing visual feedback to the operator.
robots↗
Verification of the Solar Dynamics Observatory High Gain Antenna Pointing Algorithm Using Flight Data
The Solar Dynamics Observatory (SDO) is a NASA spacecraft designed to study the Sun. It was launched on February 11, 2010 into a geosynchronous orbit, and uses a suite of attitude sensors and actuators to finely point the spacecraft at the Sun. SDO has three science instruments: the Atmospheric Imaging Assembly (AIA), the Helioseismic and Magnetic Imager (HMI), and the Extreme Ultraviolet Variability Experiment (EVE). SDO uses two High Gain Antennas (HGAs) to send science data to a dedicated ground station in White Sands, New Mexico. In order to meet the science data capture budget, the HGAs must be able to transmit data to the ground for a very large percentage of the time. Each HGA is a dual-axis antenna driven by stepper motors. Both antennas transmit data at all times, but only a single antenna is required in order to meet the transmission rate requirement. For portions of the year, one antenna or the other has an unobstructed view of the White Sands ground station. During other periods, however, the view from both antennas to the Earth is blocked for different portions of the day. During these times of blockage, the two HGAs take turns pointing to White Sands, with the other antenna pointing out to space. The HGAs handover White Sands transmission responsibilities to the unblocked antenna. There are two handover seasons per year, each lasting about 72 days, where the antennas hand off control every twelve hours. The non-tracking antenna slews back to the ground station by following a ground commanded trajectory and arrives approximately 5 minutes before the formerly tracking antenna slews away to point out into space. The SDO Attitude Control System (ACS) runs at 5 Hz, and the HGA Gimbal Control Electronics (GCE) run at 200 Hz. There are 40 opportunities for the gimbals to step each ACS cycle, with a hardware limitation of no more than one step every three GCE cycles. The ACS calculates the desired gimbal motion for tracking the ground station or for slewing, and sends the command to the GCE at 5 Hz. This command contains the number of gimbals steps for that ACS cycle, the direction of motion, the spacing of the steps, and the delay before taking the first step. The AIA and HMI instruments are sensitive to spacecraft jitter. Pre-flight analysis showed that jitter from the motion of the HGAs was a cause of concern. Three jitter mitigation techniques were developed to overcome the effects of jitter from different sources. The first method is the random step delay, which avoids gimbal steps hitting a cadence on a jitter-critical mode by pseudo-randomly delaying the first gimbal step in an ACS cycle. The second method of jitter mitigation is stagger stepping, which forbids the two antennas from taking steps during the same ACS cycle in order to avoid constructively adding jitter from two antennas. The third method is the inclusion of an instrument No Step Request (NSR), which allows the instruments to request a stoppage in gimbal stepping during the times when they are taking images. During the commissioning phase of the mission, a jitter test was performed onboard the spacecraft. Various sources of jitter, such as the reaction wheels, the High Gain Antenna motors, and the motion of the instrument filter wheels, were examined to determine the level of their effect on the instruments. During the HGA portion of the test, the jitter amplitudes from the single step of a gimbal were examined, as well as the amplitudes due to the execution of various gimbal rates. These jitter levels are compared with the gimbal jitter allocations for each instrument. Additionally, the jitter test provided insight into a readback delay that exists with the GCE. Pre-flight analysis suggested that gimbal steps scheduled to occur during the later portion of an ACS cycle would not be read during that cycle, resulting in a delay in the telemetered current gimbal position. Flight data from the jitter test confirmed this expectation. Analysis is presentehat shows the readback delay does not have a negative impact on gimbal control. The decision was made to consider implementing two of the jitter mitigation techniques on board the spacecraft: stagger stepping and the NSR. Flight data from two sets of handovers, one set without jitter mitigation and the other with mitigation enabled, were examined. The trajectory of the predicted handover was compared with the measured trajectory for the two cases, showing that tracking was not negatively impacted with the addition of the jitter mitigation techniques. Additionally, the individual gimbal steps were examined, and it was confirmed that the stagger stepping and NSRs worked as designed. An Image Quality Test was performed to determine the amount of cumulative jitter from the reaction wheels, HGAs, and instruments during various combinations of typical operations. In this paper, the flight results are examined from a test where the HGAs are following the path of a nominal handover with stagger stepping on and HMI NSRs enabled. In this case, the reaction wheels are moving at low speed and the instruments are taking pictures in their standard sequence. The flight data shows the level of jitter that the instruments see when their shutters are open. The HGA-induced jitter is well within the jitter requirement when the stagger step and NSR mitigation options are enabled. The SDO HGA pointing algorithm was designed to achieve nominal antenna pointing at the ground station, perform slews during handover season, and provide three HGA-induced jitter mitigation options without compromising pointing objectives. During the commissioning phase, flight data sets were collected to verify the HGA pointing algorithm and demonstrate its jitter mitigation capabilities.
Bourkland, Kristin L.↗
Congruent and spurious motion in the learning and performance of a compensatory tracking task.
Examination of the importance of congruent and spurious yaw motion in compensatory tracking by eight airline pilots. The pilots, seated erect in the Ames man-carrying rotation device(MCRD), tracked with k/s+1 and k/s(s+1) vehicle dynamics in fixed- and moving-base simulation. Following the learning phase of the experiment, five levels of spurious angular acceleration were superimposed on the motion of the MCRD. Learning of the tracking task was found to be a function of both vehicle dynamics and mode of simulation. The presence of congruent motion information reduced learningtime in k/s(s+1) vehicle dynamics and resulted in lower tracking error in both vehicle dynamics. The spurious angular accelerations resulted in an increase in pilot tracking error; however, the relationship between the magnitude of the acceleration and its effect was highly complex. The data suggest that the minimal disturbance level for spurious angular accelerations during tracking is below 0.4 degec.
Guercio, J. G.↗
Stereo imaging velocimetry for microgravity applications
Stereo imaging velocimetry is the quantitative measurement of three-dimensional flow fields using two sensors recording data from different vantage points. The system described in this paper, under development at NASA Lewis Research Center in Cleveland, Ohio, uses two CCD cameras placed perpendicular to one another, laser disk recorders, an image processing substation, and a 586-based computer to record data at standard NTSC video rates (30 Hertz) and reduce it offline. The flow itself is marked with seed particles, hence the fluid must be transparent. The velocimeter tracks the motion of the particles, and from these we deduce a multipoint (500 or more), quantitative map of the flow. Conceptually, the software portion of the velocimeter can be divided into distinct modules. These modules are: camera calibration, particle finding (image segmentation) and centroid location, particle overlap decomposition, particle tracking, and stereo matching. We discuss our approach to each module, and give our currently achieved speed and accuracy for each where available.
Miller, Brian B.↗
Techniques for the analysis of geodynamic effects using laser data
A technique is described by which laser measurements may be used in an explicit definition of geodetic parameters at the one-meter level of resolution. Observations made by a tracking station at the Goddard Space Flight Center of the Beacon Explorer C satellite are analyzed with this technique to yield highly precise measures of perturbations in the satellite's inclination, including the effect of earth and ocean tides and variations in the station's latitude due to polar motion. The tracking configuration is described, and the basic technique is outlined (analysis of six-hour observations with a weighted least squares orbit determination method). Techniques for analyzing orbital errors and inclination perturbations are described, and it is shown that quarter-day spans of laser data can be employed to monitor the inclination of the satellite to the order of 0.01 arcseconds precision over a period of 17 months.
Dunn, P. J.↗
Tracer particle transport dynamics in the diffusive sandpile cellular automaton
The confinement properties of the diffusive running sandpile are characterized by tracking the motion of a population of marked grains of sand. It is found that, as the relative strength of the avalanching to the diffusive transport channel is varied, a point is reached at which the particle global confinement time and the probability density functions of the jump-sizes and waiting-times of the tracked grains experience a sudden change, thus revealing a dynamical transition, that is consistent with previous studies (Newman DE et al., Phys Rev Lett 2002;88(20):204304). Across this transition, the sandpile moves from a regime characterized by self-similarity and memory, where avalanches of all possible sizes dominate transport across the system, to another regime where transport is taken over by near system-size, quasi-periodic avalanches. Values for the fractional transport exponents that quantify effective transport across the sandpile prior to the transition are also obtained.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Langley Research Center Photogrammetric Measurements of Solar Array Dynamics: Preliminary Results
The NASA Langley Research Center participated in the Solar Array Experiment with two primary objectives: (1) to study the structural and control dynamics of a new class of large, lightweight, low-frequency space structures, and (2) to develop technology for remote video measurement of structural motions. The shuttle orbiter's closed circuit television (CCTV) system was used to provide recorded video images of the solar array from four locations in the payload bay, two on the forward bulkhead and two on the aft bulkhead. White reflective targets were placed on the array to provide discrete points at which to track array motion. A dynamic test consisted of a quiescent period in which orbiter operations were inhibited and crew motion restricted, an excitation period, and a free decay period. The orbiter was placed in free drift while in a gravity gradient orientation and dynamics tests were timed to occur at orbital noon so that the Sun would illuminate one side of the array and Earth albedo the other. The CCTV system was turned on during the quiescent period, approximately 3 minutes prior to excitation. Video from all four CCTV cameras was recorded continuously through the excitation and free-decay periods for a total test record of 8 to 10 minutes.
Brumfield, M. L.↗
HARLIE Aerosol and Cloud Structure and Wind Observations during HARGLO and IHOP
The Holographic Airborne Rotating Lidar Instrument Experiment (HARLIE) is a conical-scanning, 1-micron wavelength lidar that has been adapted for ground-based applications and used to infer horizontal wind information by tracking the motions of aerosol and cloud scattering structures. In addition, HARLIE's rapid continuous scanning enables boundary layer statistics and a new cloud lidar data product that gives cloud coverage as a function of altitude with high temporal resolution. HARLIE has been used in several field campaigns while developing the techniques for wind, boundary layer (BL), and cloud data products. These campaigns involved a variety of wind measuring instruments including rawindsondes, cloud-tracked winds from video imagery, Doppler lidars and Doppler radars. HARGLO-2 was one of these campaigns and was dedicated to wind profile intercomparisons over a 1-week period in November of 2001. The most recent of these campaigns was the International H20 Project (IHOP) located in the Southern Great Plains of the US during May and June of 2002.
Schwemmer, Geary K.↗
Insights into Rotational and Translational Dynamics in Mixtures of Ethylene Glycol and Choline Chloride Using Nuclear Magnetic Resonance Techniques
This work examines molecular dynamics and interactions in ethylene glycol–choline chloride (EG–ChCl) mixtures across 0–33 mol % ChCl, spanning the true eutectic region near 17–20 mol % and the commonly used 1:2 formulation. We combine pulsed-field-gradient (PFG) diffusion, fast-field-cycling (FFC) relaxometry, temperature-dependent 13 C T 1 , and nuclear Overhauser effect spectroscopy (NOESY) to disentangle local from macroscopic dynamics. PFG and FFC show that both translational and average rotational motions largely track the strong increase in viscosity with ChCl content, with ethylene glycol consistently diffusing faster than the choline cation and no global dynamical anomaly at the eutectic composition. More subtle, site-specific composition effects nevertheless emerge. The ratio of the diffusion coefficient of the hydroxyl group of choline to the diffusion coefficient of the methyl group of choline displays a shallow minimum in the 17–25 mol % region, indicating a modest change in how the hydroxyl-bearing end of choline samples the underlying translational motion relative to the methyl groups. 13 C T 1 analysis shows that rotational correlation times at 25 °C generally increase with ChCl, reflecting viscosity-coupled slowing, while the CH 2 –N α site exhibits a small but reproducible deviation from this monotonic trend near the eutectic. NOESY spectra at similar compositions reveal enhanced cross-relaxation between EG and choline protons, consistent with increased headgroup–solvent contact density rather than a wholesale structural rearrangement. Overall, our multitechnique study demonstrates that EG–ChCl dynamics are predominantly viscosity-dominated, with the eutectic region acting as a subtle dynamical crossover where specific choline segments become maximally coupled to the hydrogen-bond network. These insights refine the structure–dynamics picture of choline-chloride DESs and provide practical guidance for tuning composition in electrochemical, separation, and catalytic applications.
diffusion↗
A System for Measuring the Sway of the Vehicle Assembly Building
A system was developed to measure the sway of the Vehicle Assembly Building (VAB) at Kennedy Space Center. This system was installed in the VAB and gathered more than one total year of data. The building movement was correlated with measurements provided by three wind towers in order to determine the maximum deflection of the building during high-wind events. The VAB owners were in the process of obtaining new platforms for use in assembling very tall rockets when analysis of the VAB showed that a high wind could move the building sufficiently that an upper platform might impact a rocket. The problem arises because safety requires a relatively small gap between the platform and the rocket, while a large enough gap is needed to ensure that stacking tolerances prevent contact between the rocket and the platform. This only leaves an inch or two (approximately 2 to 5 cm) of total clearance, so when the analysis showed that more than a couple of inches of motion could occur in a high wind, there was a potential for damaging the rocket. The KSC Applied Physics Laboratory was asked to install a system in the VAB that could measure the motion of the building in high winds to determine the actual building sway. The motion of the VAB roof under wind load was measured optically, and under analysis, it was determined that a relatively large-aperture optical system would be required to reduce diffraction effects to less than a small fraction of an inch (approximately mm) at a distance of 500 ft (≈150 m). A 10-in. (approximately 250 mm) telescope was placed on the floor of the building, looking at the ceiling. On the ceiling, a flat plate with three white LEDs was mounted in an "L" shape, such that the telescope was essentially looking at three stars. Software was written to track the motion of these three points using an image processing system. This provided a better than 1/10-in. (approximately 2.5-mm) 2D measurement faster than once a second. Data was downloaded once a month for comparison with the wind tower data. The system was fully operational and provided enough data to show that the VAB will only move 1 in. (approximately 2.5 cm) at the ceiling under 70-knot winds. Adjustable platforms are not required.
Youngquist, Robert↗
Gigatraj: An Atmospheric Trajectory Model
Atmospheric trajectory models have a long history of success in tracking air motions in the lower stratosphere and upper troposphere over periods of up to a few days. Parcels have been traced backwards from observations to identify whatever phenomena (strong convection, volcanic eruptions, rocket launches, etc.) put their signature on them. Parcels have also been initialized at a known event and traced forward to examine their subsequent physical and chemical evolution. We describe a new trajectory model, "gigatraj," that aims to increase exibility by (a) making it straightforward to use new meteorological data sources, including those not based on regular latitude-longitude grids; (b) enabling a run-time choice of vertical coordinate system for kinematic and/or quasi-isentropic calculations; (c) allowing for the output of arbitrary meteorological products, selectable by the user and interpolated to the parcels' locations and times. The model can be run in a serial or parallel processing environment, so that large numbers of parcels can be traced in a reasonable time. Information is presented on model accuracy and performance. The former is demonstrated by runs using both test-pattern winds (comparing expected paths with actual output) and real-world winds (comparing forward and backward runs to characterize how well parcels retrace their paths). Sample cases are also shown, including a reverse domain lling (RDF) calculation illustrating a tropopause fold event. Model output can be displayed using the new Visualization And Lagrangian dynamics Immersive eXtended Reality (VALIXR) system, and an example will be shown. In addition, we describe work to incorporate a version of gigatraj into the Goddard Earth Observing System (GEOS) of NASA's Global Modeling and Assimilation O ce (GMAO) at Goddard Space Flight Center. This enables trajectory calculations to be performed within the running GEOS model at the latter's native time resolution, instead of using the winds from every few hours. It also provides access to all of GEOS's internal variables as they are calculated. This module may be useful, for example, for tracking rapid chemical changes in a Lagrangian framework.
dynamics↗