VEST: Abstract vector calculus simplification in Mathematica. (English) Zbl 1344.15001
Summary: We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper [the authors, “Automation of the guiding center expansion”, Phys. Plasmas 20, Article ID 072105, 34 p. (2013; doi:10.1063/1.4813247)], we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations.
MSC:
| 15-04 | Software, source code, etc. for problems pertaining to linear algebra |
Software:
Mathematica; Cadabra; xTensor; Maple; xPerm; Invar; VEST; GeneralVectorAnalysis; TContinuumMechanics; xAct; VEST; Tensorial; MathTensor; ORTHOVEC; MathGRReferences:
| [1] | Mathematica, Version 9.0 (2012), Wolfram Research Inc.: Wolfram Research Inc. Champaign, Illinois |
| [2] | Eastwood, J. W., ORTHOVEC: version 2 of the REDUCE program for 3-D vector analysis in orthogonal curvilinear coordinates, Computer Physics Communications, 64, 1, 121-122 (1991) |
| [3] | Yasskin, P. B.; Belmonte, A., CalcLabs With Maple Commands for Multivariable Calculus (2010), Brooks/Cole, Cengage Learning: Brooks/Cole, Cengage Learning Belmont, CA |
| [5] | Wirth, M. C., Symbolic vector and dyadic analysis, SIAM Journal on Computing, 8, 3, 306 (1979) · Zbl 0434.68026 |
| [6] | Liang, S.; Jeffrey, D. J., Rule-based simplification in vector-product spaces, (Towards Mechanized Mathematical Assistants. Towards Mechanized Mathematical Assistants, Lecture Notes in Artificial Intelligence (2007)), 116-127 · Zbl 1202.68493 |
| [7] | Qin, H.; Tang, W.; Rewoldt, G., Symbolic vector analysis in plasma physics, Computer Physics Communications, 116, 107-120 (1999) · Zbl 1074.76650 |
| [9] | Cary, J. R., Lie transform perturbation theory for Hamiltonian systems, Physics Reports, 79, 2, 129-159 (1981) |
| [10] | Cary, J. R.; Brizard, A. J., Hamiltonian theory of guiding-center motion, Reviews of Modern Physics, 81, 2, 693-738 (2009) · Zbl 1205.37068 |
| [11] | Littlejohn, R. G., Variational principles of guiding centre motion, Journal of Plasma Physics, 29, 1, 111 (1983) |
| [12] | Burby, J. W.; Squire, J.; Qin, H., Automation of the guiding center expansion, Physics of Plasmas, 20, 7, 072105 (2013) |
| [14] | MacCallum, M. A.H., Computer algebra in general relativity, International Journal of Modern Physics A, 17, 20, 2707-2710 (2002) |
| [15] | Wang, Y., MathGR: a tensor and GR computation package to keep it simple |
| [16] | Parker, L.; Christensen, S., MathTensor: A System for Doing Tensor Analysis by Computer (1994), Addison-Wesley: Addison-Wesley Reading, MA |
| [17] | Peeters, K., Cadabra: a field-theory motivated symbolic computer algebra system, Computer Physics Communications, 176, 8, 550-558 (2007) · Zbl 1196.68333 |
| [20] | Stoutemyer, D., Symbolic computer vector analysis, Computers and Mathematics with Applications, 5, 1-9 (1979) · Zbl 0399.76001 |
| [21] | Martín-García, J. M., xPerm: fast index canonicalization for tensor computer algebra, Computer Physics Communications, 179, 8, 597-603 (2008) · Zbl 1197.15002 |
| [22] | Manssur, L. R.U.; Portugal, R.; Svaiter, B., Group-theoretic approach for symbolic tensor manipulation, International Journal of Modern Physics C, 13, 07, 859-879 (2002) · Zbl 1086.65515 |
| [23] | Cunningham, J., Vectors (1969), Heinemann Educational Books Ltd.: Heinemann Educational Books Ltd. London · Zbl 0174.53001 |
| [24] | Patterson, E., Solving Problems in Vector Algebra (1968), Oliver and Boyd Ltd.: Oliver and Boyd Ltd. Edinburgh, London · Zbl 0187.42601 |
| [25] | Wimmel, H. K., Extended standard vector analysis with applications to plasma physics, European Journal of Physics, 3, 4, 223 (1982) |
| [26] | Littlejohn, R. G., Geometry and guiding center motion, Contemporary Mathematics, 28, 151 (1984) · Zbl 0543.58016 |
| [27] | Lovelock, D., Dimensionally dependent identities, Mathematical Proceedings of the Cambridge Philosophical Society, 68, 2, 345 (1970) · Zbl 0197.48301 |
| [28] | Edgar, S. B.; Hoglund, A., Dimensionally dependent tensor identities by double antisymmetrization, Journal of Mathematical Physics, 43, 1, 659-677 (2002) · Zbl 1052.53022 |
| [29] | Martín-García, J. M.; Portugal, R.; Manssur, L., The invar tensor package, Computer Physics Communications, 177, 8, 640-648 (2007) · Zbl 1196.15006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
