The constrained mechanical systems in velocity component are known as nonholonomic constraints which are significantly important in engineering and robotics. A number of applicable theoretical studies have been performed on such systems among which the geometrical approach for mechanical systems has received extensive consideration. The movement direction, dynamical stability, and system control are among the topics geometrically related to mechanical (nonholonomic) systems. In this paper, a review of the geometrical point of view of mechanical systems constrained by nonholonomic constraints is represented. Moreover, we aim to find the motion equation of a ballistic missile moving towards a given target in a three-dimensional space. Initially, we calculate the motion equation of a ballistic missile which is launched towards an object moving along the z-axis with a constant velocity c. Finally, a general condition is assumed and the motion equation of the missile chasing a moving object in a R3 space along a certain curve defined by the parametrical equations x =ξ(t), y = η(t) and z = ζ(t) is calculated.
1. A. M. Bloch, and P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal. 136(1996), 21–99.
2. A. M. Bloch, Nonholonomic Mechanics and Control, Springer Verlag, New York (2003).
3. F. Cardin and M. Favreti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys. 18(1996), 295–325.
4. J. F. Carinena and M. F. Ranada, Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen. 26(1993), 1335– 1351.
5. J. Cortes, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics 1793, Springer, Berlin (2002).
6. J. Cortes, M. de Leon, J. C. Marrero and E. Martinez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst. A. 24(2009), 213–271.
7. G. Giachetta, Jet methods in nonholonomic mechanics, J. Math. Phys. 33(1992), 1652– 1655.
8. J. Janova, A Geometric theory of mechanical systems with nonholonomic constraints, Thesis, Faculty of Science, Masaryk University, Brno, 2002 (in Czech).
9. J. Janova and J. Musilova, Nonholonomic mechanics mechanics: A geometrical treatment of general coupled rolling motion, Int. J. Non-Linear Mech. 44(2009), 98–105.
10. W. S. Koon and J. E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic system, Rep. Math. Phys. 40(1997), 21–62.
11. O. Krupkova, and M. Swaczyna, Horizontal and contact forms on constraint manifods, Proc. of the 24th Winter school Geometry and Physics, Srn, 2004; Rend. Cric0 Mat. Palermo, Serie II, Suppl. 75(2005), 259–267.
12. O. Krupkova, Mechanical systems with nonholonomic constraints, J. Math. Phys. 38(1997), 5098–5126.
13. O. Krupkova, The Geometry of Ordinary Variational Equation, Springer–Verlage Berlin Heidelberg (1997).
14. O. Krupkova, The Geometry of Ordinary Differential Equations, Lecture Notes in Mathematics 1678, Springer, (1997).
15. O. Krupkova, The nonholonomic variational principle, J. Phys. A: Math. Theor. 42(2009), 185–201.
16. O. Krupkova, Geometric mechanics on nonholonomic submanifolds, Commun. Math. Sci. 18(2010), 51–77.
17. O. Krupkova and J. Musilova, The relativistic particle as a mechanical system with nonlinear constraints, J. Phys. A: Math. Gen. 34(2001), 3859–3875.
18. I. Kupka and W. M. Oliva, The nonholonomic mechanics, J. Differ. Equ. 169(2001), 169–189.
19. M. de Leon, J. C. Marrero and D. M. de Diego, Nonholonomic Lagrangian systems in jet manifolds, J. Phys. A: Math. Gen. 30(1997), 1167–1190.
20. M. de Leon, J.C. Marrero and D. M. de Diego, Mechanical systems with nonlinear constraints, Int. J. Theor. Phys. 36(1997), 979–995.
21. M. De. Leon, and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Math, Ser. 152, Amsterdam, (1989).
22. E. Massa and E. Pagani, A new look at classical mechanics of constrained systems, Ann. Inst. Henri Poincare 66(1997), 1–36.
23. P. Morando and S. Vignolo, A geometric approach to constrained mechanical systems, symmetries and inverse problems, J. Phys. A.: Math. Gen. 31(1998), 8233–8245.
24. M. F. Ranada, Time-dependent Lagrangian systems: A geometric approach to the theory of systems with constraints, J. Math. Phys. 35(1994), 748–758.
25. W. Sarlet, A direct geometrical construction of the dynamics of nonholonomic Lagrangian systems, Extr. Math. 11(1996), 202–212.
26. W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of nonholonomic Lagrangian systems, J. Phys. A: Math. Gen. 28(1995), 3253–3268.
27. D. J. Saunders, The Geometry of Jet Bundles, London Math Society Lecture Note Series, 142. Cambridge University Press, Cambridge, (1989).
28. M. Swaczyna, Examples of nonholonomic mechanical systems, Preprint Series Glob. Anal. Appl. 10(2004), 1-24.
29. M. Swaczyna, Several examples of nonholonomic mechanical systems, Comm. Math. 19(2011), 27–56.
Azizpour, E., & Moazzami, G. (2022). Tracking a target in a three-dimensional space by a nonholonomic constraint. Journal of Finsler Geometry and its Applications, 3(1), 72-85. doi: 10.22098/jfga.2022.10560.1065
MLA
Esmaeil Azizpour; Ghazale Moazzami. "Tracking a target in a three-dimensional space by a nonholonomic constraint", Journal of Finsler Geometry and its Applications, 3, 1, 2022, 72-85. doi: 10.22098/jfga.2022.10560.1065
HARVARD
Azizpour, E., Moazzami, G. (2022). 'Tracking a target in a three-dimensional space by a nonholonomic constraint', Journal of Finsler Geometry and its Applications, 3(1), pp. 72-85. doi: 10.22098/jfga.2022.10560.1065
VANCOUVER
Azizpour, E., Moazzami, G. Tracking a target in a three-dimensional space by a nonholonomic constraint. Journal of Finsler Geometry and its Applications, 2022; 3(1): 72-85. doi: 10.22098/jfga.2022.10560.1065
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