From a dynamical system of the knee to natural jet geometrical objects
Authors:
Mircea Neagu and Mihaela Maria Marin
Journal:
Quart. Appl. Math. 71 (2013), 689-705
MSC (2000):
Primary 58A20, 53C07; Secondary 53C43, 83C22.
DOI:
https://doi.org/10.1090/S0033-569X-2013-01307-4
Published electronically:
September 5, 2013
MathSciNet review:
3136991
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, we construct some natural geometrical objects on the $1$-jet space $J^{1}(\mathbb {R},\mathbb {R}^{3})$, like a nonlinear connection, a Cartan linear connection (together with its d-torsions and d-curvatures), a jet “electromagnetic" d-field and its geometric “electromagnetic" Yang-Mills energy, starting from a given dynamical system governing the three-dimensional motion of the knee in the mathematical model introduced by Grood and Suntay. The computer-drawn graphics of corresponding Yang-Mills energetic surfaces of constant level (produced by this knee dynamical system) are given.
References
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- D. M. Barbu, I. Barbu, Dynamical model for an original mechatronical rehabilitation system, Proceedings of the 14th WSEAS International Conference on Applied Mathematics (2009), 23-26.
- E. S. Grood, W. J. Suntay, A joint coordinate system for the clinical description of three-dimensional motions: application to the knee, ASME, J. Biomechanical Eng., vol. 105 (1983), 136-144.
- M. S. Hefzy, E. M. Abdel-Rahman, Three-dimensional dynamic anatomical modeling of the human knee joint, in “Biomechanical Systems. Techniques and Applications”, Vol. III - “Musculoskeletal Models and Techniques” (Cornelius Leondes, ed.), CRC Press, 2000.
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- Radu Miron and Mihai Anastasiei, The geometry of Lagrange spaces: theory and applications, Fundamental Theories of Physics, vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1281613
- Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734
- I. C. Roşca, C. Radu, On the design of the analytical ankle-foot model used to determine dynamic parameters during locomotion, Online Proceedings of the 6th International DAAAM Baltic Conference “Industrial Engineering. I. Design Engineering” (2008), Tallinn, Estonia.
- D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series, vol. 142, Cambridge University Press, Cambridge, 1989. MR 989588
- Constantin Udrişte, Geometric dynamics, Mathematics and its Applications, vol. 513, Kluwer Academic Publishers, Dordrecht, 2000. Chapter 11 by Lucia Drăguşin. MR 1778560
References
- J. Apkarian, S. Naumann, B. Cairns, A three-dimensional kinematic and dynamic model for the lower limb, J. Biomechanics, vol. 22, no. 2 (1989), 143-155.
- G. S. Asanov, Jet extension of Finslerian gauge approach, Fortschr. Phys. 38 (1990), no. 8, 571–610. MR 1076500 (91k:83063), DOI https://doi.org/10.1002/prop.2190380802
- Vladimir Balan and Mircea Neagu, Jet Single-Time Lagrange Geometry and Its Applications, John Wiley and Sons Inc., Hoboken, NJ, 2011. MR 2895129
- D. M. Barbu, I. Barbu, Dynamical model for an original mechatronical rehabilitation system, Proceedings of the 14th WSEAS International Conference on Applied Mathematics (2009), 23-26.
- E. S. Grood, W. J. Suntay, A joint coordinate system for the clinical description of three-dimensional motions: application to the knee, ASME, J. Biomechanical Eng., vol. 105 (1983), 136-144.
- M. S. Hefzy, E. M. Abdel-Rahman, Three-dimensional dynamic anatomical modeling of the human knee joint, in “Biomechanical Systems. Techniques and Applications”, Vol. III - “Musculoskeletal Models and Techniques” (Cornelius Leondes, ed.), CRC Press, 2000.
- V. G. Ivancevic, New mechanics of generic musculo-skeletal injury, http://arXiv.org/q-bio.TO/0807.1759v5 (2009).
- J. Y. S. Luh, M. W. Walker, and R. P. C. Paul, On-line computational scheme for mechanical manipulators, Trans. ASME Ser. G J. Dynamic Systems Measurement Control 102 (1980), no. 2, 69–76. MR 586929 (81g:70001)
- Radu Miron and Mihai Anastasiei, The geometry of Lagrange spaces: theory and applications, Fundamental Theories of Physics, vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1281613 (95f:53120)
- Peter J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734 (88f:58161)
- I. C. Roşca, C. Radu, On the design of the analytical ankle-foot model used to determine dynamic parameters during locomotion, Online Proceedings of the 6th International DAAAM Baltic Conference “Industrial Engineering. I. Design Engineering” (2008), Tallinn, Estonia.
- D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, vol. 142, Cambridge University Press, Cambridge, 1989. MR 989588 (90f:58007)
- C. Udrişte, Geometric Dynamics, Kluwer Academic Publishers, 2000. MR 1778560 (2001k:70016)
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Additional Information
Mircea Neagu
Address at time of publication:
University Transilvania of Braşov, Department of Mathematics-Informatics, Blvd. Iuliu Maniu, no. 50, Braşov 500091, Romania.
Email:
mircea.neagu@unitbv.ro
Mihaela Maria Marin
Address at time of publication:
University Transilvania of Braşov, Faculty of Mechanical Engineering, Str. Politehnicii, no. 1, Braşov 500024, Romania
Email:
marin_mihaela_maria@yahoo.com
Keywords:
1-jet spaces,
dynamical system of knee,
jet least squares Lagrangian function,
jet single-time Lagrange geometry,
“electromagnetic” Yang-Mills energy.
Received by editor(s):
November 16, 2011
Published electronically:
September 5, 2013
Article copyright:
© Copyright 2013
Brown University
The copyright for this article reverts to public domain 28 years after publication.
