Authors

V. N. Salnikov, A. Hamdouni

Abstract

In this paper, we discuss the possibility of using computer algebra tools in the process of modeling and qualitative analysis of mechanical systems and problems from theoretical physics. We describe some constructions—Courant algebroids and Dirac structures—from the so-called generalized geometry. They prove to be a convenient language for studying the internal structure of the differential equations of port-Hamiltonian and implicit Lagrangian systems, which describe dissipative or coupled mechanical systems and systems with constraints, respectively. For both classes of systems, we formulate some open problems that can be solved using computer algebra tools and methods. We also recall the definitions of graded manifolds and Q ‑structures from graded geometry. On particular examples, we explain how classical differential geometry is described in the framework of the graded formalism and what related computational questions can arise. This direction of research is apparently an almost unexplored branch of computer algebra.

Citation

  • Journal: Programming and Computer Software
  • Year: 2020
  • Volume: 46
  • Issue: 2
  • Pages: 126–132
  • Publisher: Pleiades Publishing Ltd
  • DOI: 10.1134/s0361768820020097

BibTeX

@article{Salnikov_2020,
  title={{Differential Geometry and Mechanics: A Source for Computer Algebra Problems}},
  volume={46},
  ISSN={1608-3261},
  DOI={10.1134/s0361768820020097},
  number={2},
  journal={Programming and Computer Software},
  publisher={Pleiades Publishing Ltd},
  author={Salnikov, V. N. and Hamdouni, A.},
  year={2020},
  pages={126--132}
}

Download the bib file

References

  • Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
  • Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics vol. 57 133–156 (2006)10.1016/j.geomphys.2006.02.009
  • Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part II: Variational structures. Journal of Geometry and Physics vol. 57 209–250 (2006) – 10.1016/j.geomphys.2006.02.012
  • Salnikov, V. & Hamdouni, A. From modelling of systems with constraints to generalized geometry and back to numerics. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik vol. 99 (2019) – 10.1002/zamm.201800218
  • Razafindralandy, D., Salnikov, V., Hamdouni, A. & Deeb, A. Some robust integrators for large time dynamics. Advanced Modeling and Simulation in Engineering Sciences vol. 6 (2019) – 10.1186/s40323-019-0130-2
  • Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
  • van der Schaft, A., Port-Hamiltonian systems: An introductory survey, Proc. Int. Congr. Mathematicians, Madrid, 2006.
  • W.M. Tulczyjew. Tulczyjew, W.M., The Legendre transformation, Ann. Inst. H. Poincaré,Sect. A, 1977, vol. 27, no. 1, pp. 101–114. (1977)
  • Verlet, L. Computer ‘Experiments’ on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review vol. 159 98–103 (1967) – 10.1103/physrev.159.98
  • Yoshida, H. Construction of higher order symplectic integrators. Physics Letters A vol. 150 262–268 (1990) – 10.1016/0375-9601(90)90092-3
  • A. Falaize. Falaize, A. and Hélie, T., Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes piano, J. Sound Vib., 2016. (2016)
  • Kotov, A., Schaller, P. & Strobl, T. Dirac Sigma Models. Communications in Mathematical Physics vol. 260 455–480 (2005) – 10.1007/s00220-005-1416-4
  • Salnikov, V. and Hamdouni, A., Geometric integrators in mechanics: The need for computer algebra tools, Proc. 3rd Int. Conf. Computer Algebra, Moscow, 2019.
  • Salnikov, V. and Hamdouni, A., Géométrie généralisée et graduée pour la mécanique, Proc. Congrès Français de Mécanique, Brest, France, 2019.
  • A. Kushner. Kushner, A., Lychagin, V., and Rubtsov, V., Contact geometry and non-linear differential equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007. (2007)
  • Krasil’ shchik, I. S. Higher symmetries and conservation laws. Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations 57–97 (2000) doi:10.1007/978-94-017-3196-6_2 – 10.1007/978-94-017-3196-6_2
  • Hamdouni, A. and Salnikov, V., Dirac integrators for port-Hamiltonian systems, in prep.
  • Salnikov, V. and Hamdouni, A., Discretization in the graded world, in prep.