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All sourcesNASA NTRSDOE OSTI
NASA NTRS2002

Science Planning for Multi-Spacecraft Coordinated Observations

Fulfilling the promise of an era of great observatories, NASA now has more than three space-based astronomical telescopes operating in different wavebands. This situation provides astronomers with a unique opportunity to simultaneously observe with multiple observatories. Yet scheduling multiple observatories simultaneously is highly inefficient when compared to single observatory observations. Thus, programs using multiple observatories are limited not due to scientific restrictions, but due to operational inefficiencies. Each year, a number of proposals are accepted by a space-based observatory for conduction of astronomical observations and gathering of science data for the study of galactic events. Since each space-based observatory uses a set of instruments designed to operate in specific energy regions, most such studies are conducted by submitting observation proposals to multiple observatories, with requests to coordinate among themselves. To assure that the proposed observations can be scheduled, each observatory's staff has to check that the observations are valid and meet all the constraints for their own observatory; in addition, they have to verify that the observations satisfy the constraints of the other observatories. Thus, coordinated observations require painstaking manual collaboration among the observatory staff at each observatory. In order to exploit new paradigms for observatory operation, the Goddard Space Flight Center's Advanced Architectures and Automation Branch has developed a prototype tool called the Visual Observation Layout Tool (VOLT). The main objective of VOLT is to provide a visual tool to automate the science planning of coordinated observations for multiple spacecraft, as well as to increase the scheduling probability of observations. However, VOLT is also useful for single observatory planning to optimize observatory control. Three space-based missions are interested in using VOLT (the Hubble Space Telescope, the Chandra X-Ray Observatory, and the Far Ultraviolet Spectroscopic Explorer). The VOLT team members have collaborated with these missions to gather requirements and obtain feedback on their mission planning processes. VOLT has been developed as a cross-platform Java client application for use by scientists and observatory science planning staff to visualize scheduling options and constraints. It also supports a lightweight graphical user interface for remote viewing via a Web front end. Additionally, it uniquely supports the ability to interact with multiple, diverse scheduling packages in order to determine windows of opportunity for observations and visually portray the constraints of each observation request. VOLT enables science data capture scenarios which are currently either impossible, or which require extensive time and manpower to coordinate amongst multiple observatories. it supports early detection of planning conflicts by generating coordinated solutions based on observatory schedulability and constraints. The project development approach has included frequent prototype demonstrations to our interested missions to obtain feedback after each release of the software. We will present an overview of our lessons learned in infusing the VOLT tool into the operations of the missions we have collaborated with and a brief demonstration of the software.

Maks, Lori↗
NASA NTRS2003

Coordinated turn-and-reach movements. II. Planning in an external frame of reference

The preceding study demonstrated that normal subjects compensate for the additional interaction torques generated when a reaching movement is made during voluntary trunk rotation. The present paper assesses the influence of trunk rotation on finger trajectories and on interjoint coordination and determines whether simultaneous turn-and-reach movements are most simply described relative to a trunk-based or an external reference frame. Subjects reached to targets requiring different extents of arm joint and trunk rotation at a natural pace and quickly in normal lighting and in total darkness. We first examined whether the larger interaction torques generated during rapid turn-and-reach movements perturb finger trajectories and interjoint coordination and whether visual feedback plays a role in compensating for these torques. These issues were addressed using generalized Procrustes analysis (GPA), which attempts to overlap a group of configurations (e.g., joint trajectories) through translations and rotations in multi-dimensional space. We first used GPA to identify the mean intrinsic patterns of finger and joint trajectories (i.e., their average shape irrespective of location and orientation variability in the external and joint workspaces) from turn-and-reach movements performed in each experimental condition and then calculated their curvatures. We then quantified the discrepancy between each finger or joint trajectory and the intrinsic pattern both after GPA was applied individually to trajectories from a pair of experimental conditions and after GPA was applied to the same trajectories pooled together. For several subjects, joint trajectories but not finger trajectories were more curved in fast than slow movements. The curvature of both joint and finger trajectories of turn-and-reach movements was relatively unaffected by the vision conditions. Pooling across speed conditions significantly increased the discrepancy between joint but not finger trajectories for most subjects, indicating that subjects used different patterns of interjoint coordination in slow and fast movements while nevertheless preserving the shape of their finger trajectory. Higher movement speeds did not disrupt the arm joint rotations despite the larger interaction torques generated. Rather, subjects used the redundant degrees of freedom of the arm/trunk system to achieve similar finger trajectories with differing joint configurations. We examined finger movement patterns and velocity profiles to determine the frame of reference in which turn-and-reach movements could be most simply described. Finger trajectories of turn-and-reach movements had much larger curvatures and their velocity profiles were less smooth and less bell-like in trunk-based coordinates than in external coordinates. Taken together, these results support the conclusion that turn-and-reach movements are controlled in an external frame of reference.

NASA Discipline Neuroscience↗
NASA NTRS2004

Autonomous Multi-Sensor Coordination: The Science Goal Monitor

Many dramatic earth phenomena are dynamic and coupled. In order to fully understand them, we need to obtain timely coordinated multi-sensor observations from widely dispersed instruments. Such a dynamic observing system must include the ability to Schedule flexibly and react autonomously to sciencehser driven events; Understand higher-level goals of a sciencehser defined campaign; Coordinate various space-based and ground-based resources/sensors effectively and efficiently to achieve goals. In order to capture transient events, such a 'sensor web' system must have an automated reactive capability built into its scientific operations. To do this, we must overcome a number of challenges inherent in infusing autonomy. The Science Goal Monitor (SGM) is a prototype software tool being developed to explore the nature of automation necessary to enable dynamic observing. The tools being developed in SGM improve our ability to autonomously monitor multiple independent sensors and coordinate reactions to better observe dynamic phenomena. The SGM system enables users to specify what to look for and how to react in descriptive rather than technical terms. The system monitors streams of data to identify occurrences of the key events previously specified by the scientisther. When an event occurs, the system autonomously coordinates the execution of the users' desired reactions between different sensors. The information can be used to rapidly respond to a variety of fast temporal events. Investigators will no longer have to rely on after-the-fact data analysis to determine what happened. Our paper describes a series of prototype demonstrations that we have developed using SGM and NASA's Earth Observing-1 (EO-1) satellite and Earth Observing Systems' Aqua/Terra spacecrafts' MODIS instrument. Our demonstrations show the promise of coordinating data from different sources, analyzing the data for a relevant event, autonomously updating and rapidly obtaining a follow-on relevant image. SGM was used to investigate forest fires, floods and volcanic eruptions. We are now identifying new Earth science scenarios that will have more complex SGM reasoning. By developing and testing a prototype in an operational environment, we are also establishing and gathering metrics to gauge the success of automating science campaigns.

Koratkar, Anuradha↗
NASA NTRS1997

Locomotor head-trunk coordination strategies following space flight

During locomotion, angular head movements act in a compensatory fashion to oppose the vertical trunk translation that occurs during each step in the gait cycle. This coordinated strategy between head and trunk motion serves to aid gaze stabilization and perhaps simplifies the sensory coordinate transformation between the head and trunk, allowing efficient descending motor control during locomotion. Following space flight, astronauts often experience oscillopsia during locomotion in addition to postural and gait instabilities, suggesting a possible breakdown in head-trunk coordination. The goal of the present investigation was to determine if exposure to the microgravity environment of space flight induces alteration in head-trunk coordination during locomotion. Astronaut subjects were asked to walk (6.4 km/h, 20 s trials) on a motorized treadmill while visually fixating on a centrally located earthfixed target positioned either 2 m (FAR) or 30 cm (NEAR) from the eyes. In addition, some trials were also performed during periodic visual occlusion. Head and trunk kinematics during locomotion were determined with the aid of a video-based motion analyzing system. We report data collected preflight (10 days prior to launch) and postflight (2 to 4 hours after landing). The coherence between pitch head and vertical trunk movements during gaze fixation of both FAR and NEAR targets was significantly reduced following space flight indicating decreased coordination between the head and trunk during postflight locomotion. Astronauts flying on their first mission showed greater alterations in the frequency spectra of pitch head movements as compared to their more experienced counterparts. These modifications in the efficacy of head movement control may account for the reported disruption in gaze performance during locomotion and may contribute to postflight postural and gait dysfunction.

NASA Discipline Neuroscience↗
NASA NTRS2005

Optimal Sampling of a Reaction Coordinate in Molecular Dynamics

Estimating how free energy changes with the state of a system is a central goal in applications of statistical mechanics to problems of chemical or biological interest. From these free energy changes it is possible, for example, to establish which states of the system are stable, what are their probabilities and how the equilibria between these states are influenced by external conditions. Free energies are also of great utility in determining kinetics of transitions between different states. A variety of methods have been developed to compute free energies of condensed phase systems. Here, I will focus on one class of methods - those that allow for calculating free energy changes along one or several generalized coordinates in the system, often called reaction coordinates or order parameters . Considering that in almost all cases of practical interest a significant computational effort is required to determine free energy changes along such coordinates it is hardly surprising that efficiencies of different methods are of great concern. In most cases, the main difficulty is associated with its shape along the reaction coordinate. If the free energy changes markedly along this coordinate Boltzmann sampling of its different values becomes highly non-uniform. This, in turn, may have considerable, detrimental effect on the performance of many methods for calculating free energies.

Pohorille, Andrew↗
NASA NTRS2018

Applying the SOFIA Coordinate System to the HIRMES Instrument

The High-resolution Mid-infrared Spectrometer (HIRMES) will be in-flight aboard the Stratospheric Observatory for Infrared Astronomy (SOFIA) in late 2019, which will allow for observations of the structure and evolution of protoplanetary disks. SOFIA has a different coordinate system from the system being used to build HIRMES and needs to be defined and applied to HIRMES. This is necessary because the first mirror, which is aiming the incoming light onto the slit wheel by allowing for tip and tilt adjustments, cannot rotate. Thus, the SOFIA coordinate system allows for this additional degree of freedom and allows for the slit wheel to be in line with the telescope. Using a laser radar, multiple measurements around the instrument were taken of tooling balls, which quantified an uncertainty with the measurements. Also, a laser radar was used to scan the entirety of the instrument. The center of the front flange was determined using the measurements and scans of the instrument, which was used to determine the origin of the SOFIA coordinate system and its uncertainty. By knowing the center of the front flange, an off-center, rotated coordinate system was created, matching the mechanics' schematics for the system. Going forward, this coordinate system will allow for continued alignment of the instrument in preparation for flight and for accurate measurements when aboard SOFIA.

Wraback, Elizabeth↗
DOE OSTI2014

Coordinating Garbage Collection for Arrays of Solid-state Drives

Although solid-state drives (SSDs) offer significant performance improvements over hard disk drives (HDDs) for a number of workloads, they can exhibit substantial variance in request latency and throughput as a result of garbage collection (GC). When GC conflicts with an I/O stream, the stream can make no forward progress until the GC cycle completes. GC cycles are scheduled by logic internal to the SSD based on several factors such as the pattern, frequency, and volume of write requests. When SSDs are used in a RAID with currently available technology, the lack of coordination of the SSD-local GC cycles amplifies this performance variance. We propose a global garbage collection (GGC) mechanism to improve response times and reduce performance variability for a RAID of SSDs. We include a high-level design of SSD-aware RAID controller and GGC-capable SSD devices and algorithms to coordinate the GGC cycles. We develop reactive and proactive GC coordination algorithms and evaluate their I/O performance and block erase counts for various workloads. Our simulations show that GC coordination by a reactive scheme improves average response time and reduces performance variability for a wide variety of enterprise workloads. For bursty, write-dominated workloads, response time was improved by 69% and performance variability was reduced by 71%. We show that a proactive GC coordination algorithm can further improve the I/O response times by up to 9% and the performance variability by up to 15%. We also observe that it could increase the lifetimes of SSDs with some workloads (e.g. Financial) by reducing the number of block erase counts by up to 79% relative to a reactive algorithm for write-dominant enterprise workloads.

Flash Memory↗
DOE OSTI2018

Positioning People for Effective Communication and Coordination - 18033

If the aim of people working together is to solve complex problems and develop creative solutions, would it make sense to position people to actually do this kind of work? A growing number of project teams span organizational boundaries: government agencies coordinate with other government agencies, private companies conduct major projects for government agencies, and public utilities coordinate decision-making with citizen groups. In such instances, these project teams are especially vulnerable to experiencing a gap in expectations in terms of how people should communicate and coordinate with one another. If these gaps are not addressed, communication bogs down, those involved in the project become frustrated and impatient, and solving complex problems and developing creative solutions becomes challenging if not impossible. However, it is possible to position people to effectively communicate and coordinate. One approach is to give attention to the discussion and development of a P3 Plan that describes how the project team agrees to coordinate its efforts and toward what ends. A P3 Plan is developed by the project team typically through one or more facilitated conversations. Multiple US government-to-government, government/contractor, and government/public citizen groups and teams have used this approach and have been successful moving beyond their own self-interests to deal with controversial issues. The project benefits because the work product combines the best thinking from everyone, eliminating potential political and turf battles from the start. (authors)

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2017

Coordination-Supported Imidazolate Networks: Water- and Heat-Stable Mesoporous Polymers for Catalysis

The poor water stability of most porous coordination polymers (PCPs) or metal-organic frameworks (MOFs) is widely recognised as a barrier hampering their practical applications. Herein, a facile and scalable route to prepare metal-containing polymers with a good stability in boiling water (100°C, 24 h) and air (up to 390°C) is presented. The bifunctional 1-vinylimidazole (VIm) with both a coordinating site and a polymerizable organic group is introduced as the building block. This core strategy includes the synthesis of a rigid monomer with four VIm branches via a coordination process at room temperature, followed by a radical polymerization. Here we call this material Coordination-supported Imidazolate Networks (CINs). Interestingly, CINs are composed of rich mesopores from 2 to 15 nm, as characterized by low-energy (60 kV) STEM-HAADF images. Especially, the stable CINs illustrate a high turnover frequency (TOF) of 779 h -1 in the catalytic oxidation of phenol with H 2 O as the green solvent.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Coordinate transformation and construction of finite element mesh in a diverted tokamak geometry

A coordinate transformation technique between straight magnetic field line coordinate system (Ψ, θ) and Cartesian coordinate system (R, Z) is presented employing a Solov'ev solution of the Grad-Shafranov equation. Employing the equilibrium solution, the poloidal magnetic flux Ψ(R, Z) of a diverted tokamak, magnetic field line equation is solved computationally to find curves of constant poloidal angle θ, which provides us with explicit relations R = R(Ψ, θ) and Z = Z(Ψ, θ). Correspondingly, conversion from one coordinate to the other along particle trajectories in the vicinity of separatrix is demonstrated. Based on the magnetic structure, a finite element mesh is generated in a diverted tokamak geometry to solve Poisson's equation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Controlling Cation-Cation Interactions in Uranyl Coordination Dimers by Varying the Length of the Dicarboxylate Linker

The chemistry of linear uranyl(V/VI) dioxo cations, [O yl –U–O yl ] +/2+ , is dominated by coordination of uranium in the equatorial plane. Effects of this constraint were evaluated by experiment and theory for gas-phase mixed-valence U V/VI coordination dimers in which uranyl moieties are linked by alkyl dicarboxylates, [(UO 2 + )(UO 2 2+ )(OOC-(CH 2 ) n-2 -COO 2– ) 2 ] – (n = 3–12). Faster O 2 -addition to dimers with short linkers n = 3 and 4, vs. n ≥ 5, suggests a structural difference. Computed structures with the shortest linkers have bridging dicarboxylates and nearly parallel, non-interacting uranyls. Longer linkers, n = 5–7, accommodate uranyl orientations with distinct U V –U VI end-on cation-cation interactions (CCIs), whereby Lewis base O yl from U V coordinates to the acid U VI , denoted as U V O yl ···U VI . The dimer structure for n = 8 has a U V –U VI side-on diamond-shape CCI, with U V O yl ···U VI and U VI O yl ···U V interactions. Addition of O 2 to the n = 4 and 5 dimers yields [(UO22+)2(OOC-(CH2)n-2-COO2–)2(O2–)]–, with U V oxidized to U VI and O 2 reduced to O 2 – . Whereas O 2 can associate to and oxidize the exposed U V center for dimers with n = 3 and 4, the more crowded U V site in the CCI structures inhibits O 2 addition. Finally, the results demonstrate rational structural control of uranyl-uranyl bonding and reactivity in small coordination complexes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Determine the coordination of Al and Si of Fe-rich fly ash: The challenge in NMR and how we probe its heterogeneity using X-ray spectromicroscopy

Scanning transmission X-ray microscopy (STXM) coupled with ptychographic imaging at Al and Si K-edge is a new probe to the microscale chemistry of heterogenous materials. Here, the combined techniques are applied to study local coordination environment of Fe-rich fly ash (FA). Ptychographic imaging at Al K- and Si K-edge highlights the fine Al- and Si-rich morphological features, respectively, at 6 nm pixel resolution. STXM visualizes the inter- and intra-particle variations in the silicate polymerization degree and two types of Al coordination environment in FA. The Si K-edge of SiO4 connected with four-fold coordinated Al is higher than SiO4 sharing oxygen with six-fold coordinated Al. The result is consistent with bulk nuclear magnetic resonance spectroscopy measurement of the Fe-rich sample which, however, is sensitive to paramagnetic Fe and time-consuming. Here, we demonstrate that the combined method has great potential in the studies of chemically heterogenous aluminosilicates.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2021

Decisions and coordination of retailer-led low-carbon supply chain under altruistic preference

A low-carbon supply chain formed by a dominant retailer and a small and medium-sized manufacturer (SMM) is considered. Because the SMM faces the high cost pressure of adopting carbon emission reduction (CER) technologies, the retailer may take an altruistic preference for the long-term sustainability of the chain. Three decision-making models, centralized, decentralized without altruistic preference, and decentralized with altruistic preference, are constructed to compare decisions and profits of both parties. Since neither decentrailized models can lead to a coordinated solution that is incentive compitable, a coordination contract, referred to as the cost sharing contract with altruistic preference, is proposed. Numerical study shows that the altruistic preference can help increase the SMM's profit and system efficiency but decrease the retailer's profit. The coordination contract requires the retailer to adjust the unit profit and share more than half of the CER cost. Finally, it is found that the wholesale price is the lowest in the coordination contract and the comparison of unit profit depends on the coefficient of CER cost.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2018

Investigation of a “coupling model” of coordination between low-carbon development and urbanization in China

Based on data from 30 provinces in China, this paper builds evaluation models or a carbon emission-urbanization system that explores how to achieve low-carbon development during a rapid urbanization phase. Through mathematical and statistical analysis, principal component analysis and entropy evaluation methods are applied to calculate index weights for a comprehensive evaluation index of carbon emission-urbanization. A coordination degree model and a coupling coordination degree model (CCDM) for a carbon emission-urbanization system (CUS) that explores how to achieve low-carbon development during a rapid urbanization phase. Scenario analyses and case study were applied to illustrate the results, which show five basic conclusions. (1) Low-carbon development doesn't require eliminating energy consumption completely during urbanization (2) The average level of urbanization is relatively low owing to the large disparities among provinces and the provinces’ economic development. Though the development of low-carbon in the 30 provinces is generally rapid, the gap between the highest provinces and the lowest provinces is relatively large because of their different socio-economic features. (3) Much more attention should be paid to CO 2 emissions per capita. The quality of public social service,the basic and medical insurance coverage for the elderly, the community service coverage in infrastructure and the green design during urban construction should be improved. (4) The coordination of CUS is closely related to the different development stages and geographic locations of each province. (5) For different types of provinces with different degrees of coupling coordination,there is a need to explore different development directions.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2018

The international implications of national and local coordination on building energy codes: Case studies in six cities

Building energy efficiency is an important strategy for reducing greenhouse gas emissions globally. In fact, 55 countries have included building energy efficiency in their Nationally Determined Contributions (NDCs) under the Paris Agreement. This research uses building energy code implementation in six cities across different continents as case studies to assess what it may take for countries to implement the ambitions of their energy efficiency goals. Specifically, we look at the cases of Bogota, Colombia; Da Nang, Vietnam; Eskisehir, Turkey; Mexico City, Mexico; Rajkot, India; and Tshwane, South Africa, all of which are “deep dive” cities under the Sustainable Energy for All's Building Efficiency Accelerator. The research focuses on understanding the baseline with existing gaps in implementation and coordination. The methodology used a combination of surveys on code status and interviews with stakeholders at the local and national level, as well as review of published documents. We looked at code development, implementation, and evaluation. The cities are all working to improve implementation, however, the challenges they currently face include gaps in resources, capacity, tools, and institutions to check for compliance. Better coordination between national and local governments could help improve implementation, but that coordination is not yet well established. For example, all six of the cities reported that there was little to no involvement of local stakeholders in development of the national code; only one city reported that it had access to national funding to support code implementation. More robust coordination could better link cities with capacity building and funding for compliance, and ensure that the code reflects local priorities. By understanding gaps in implementation, it can also help in designing more targeted interventions to scale up energy savings.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2014

Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates

In applications such as astrophysics and inertial confinement fusion, there are many three-dimensional cylindrical-symmetric multi-material problems which are usually simulated by Lagrangian schemes in the two-dimensional cylindrical coordinates. For this type of simulation, a critical issue for the schemes is to keep spherical symmetry in the cylindrical coordinate system if the original physical problem has this symmetry. In the past decades, several Lagrangian schemes with such symmetry property have been developed, but all of them are only first order accurate. In this paper, we develop a second order cell-centered Lagrangian scheme for solving compressible Euler equations in cylindrical coordinates, based on the control volume discretizations, which is designed to have uniformly second order accuracy and capability to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. The scheme maintains several good properties such as conservation for mass, momentum and total energy, and the geometric conservation law. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of accuracy, symmetry, non-oscillation and robustness. The advantage of higher order accuracy is demonstrated in these examples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2012

N-(sulfoethyl) iminodiacetic acid-based lanthanide coordination polymers: Synthesis, magnetism and quantum Monte Carlo studies

A series of lanthanide coordination polymers have been obtained through the hydrothermal reaction of N-(sulfoethyl) iminodiacetic acid (H{sub 3}SIDA) and Ln(NO{sub 3}){sub 3} (Ln=La, 1; Pr, 2; Nd, 3; Gd, 4). Crystal structure analysis exhibits that lanthanide ions affect the coordination number, bond length and dimension of compounds 1-4, which reveal that their structure diversity can be attributed to the effect of lanthanide contraction. Furthermore, the combination of magnetic measure with quantum Monte Carlo(QMC) studies exhibits that the coupling parameters between two adjacent Gd{sup 3+} ions for anti-anti and syn-anti carboxylate bridges are -1.0 Multiplication-Sign 10{sup -3} and -5.0 Multiplication-Sign 10{sup -3} cm{sup -1}, respectively, which reveals weak antiferromagnetic interaction in 4. - Graphical abstract: Four lanthanide coordination polymers with N-(sulfoethyl) iminodiacetic acid were obtained under hydrothermal condition and reveal the weak antiferromagnetic coupling between two Gd{sup 3+} ions by Quantum Monte Carlo studies. Highlights: Black-Right-Pointing-Pointer Four lanthanide coordination polymers of H{sub 3}SIDA ligand were obtained. Black-Right-Pointing-Pointer Lanthanide ions play an important role in their structural diversity. Black-Right-Pointing-Pointer Magnetic measure exhibits that compound 4 features antiferromagnetic property. Black-Right-Pointing-Pointer Quantum Monte Carlo studies reveal the coupling parameters of two Gd{sup 3+} ions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Quantum mechanics with coordinate dependent noncommutativity

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacobi identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗