Abstract
Methods for constructing systems of second-order ordinary differential equations whose solutions satisfy the constraint stabilization conditions are outlined. Modified Helmholtz conditions are used to bring the system to the form of the Lagrange equations. The expression of the central force is determined, which ensures the stabilization of the points movement trajectory.
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REFERENCES
A. S. Galiullin, Methods of Solving Inverse Problems of Dynamics (Nauka, Moscow, 1986) [in Russian].
R. G. Mukharlyamov, ‘‘On the construction of a set of systems of differential equations of stable motion on an integral manifold,’’ Differ. Equat. 5, 688–699 (1969).
J. Baumgarte, ‘‘Stabilization of constraints and integrals of motion in dynamical systems,’’ Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972). https://doi.org/10.1016/0045-7825(72)90018-7.
R. M. Santilli, Foundations of Theoretical Mechanics. I: The Inverse Problemin Newtonian Mechanics (Springer, New York, 1978).
R. M. Santilli, Foundations of Theoretical Mechanics. II: Birkhoffian Generalization of Hamiltonian Mechanics (Springer, New York, 1982).
N. P. Erugin, ‘‘Construction of the entire set of systems of differential equations having a given integral curve,’’ Prikl. Mat. Mech. 21, 659–670 (1952).
N. P. Erugin, A Book for Reading on the General Course of Differential Equations (Nauka Tekhnol., Minsk, 1979).
R. G. Mukharlyamov, ‘‘On the construction of systems of differential equations of motion of mechanical systems,’’ Differ. Equat. 39, 343–353 (2003).
U. Ascher, H. Chin, and S. Reich, ‘‘Stabilization of DAEs and invariant manifolds,’’ Numer. Math. 67, 131–149 (1994).
G. Kielau and P. Maisser, ‘‘A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian,’’ Z. Angew. Math. Mech. 86, 722–735 (2006).
I. E. Kaspirovich and R. G. Mukharlyamov, ‘‘On methods for constructing dynamic equations with regard for constraint stabilization,’’ Mech. Solids 54, 589–597 (2019).
R. G. Mukharlyamov, ‘‘Control of program motion of an adaptive optical system,’’ Vestn. RUDN, Ser. Prikl. Mat. Komp’yut. Nauki, No. 1, 22–40 (1994).
I. Newton, Mathematical Principles of Natural Philosophy, Series: Classics of Science (Nauka, Moscow, 1989) [in Russian].
M. G. Bertrand, ‘‘Theoreme relatife au mouvement d’un point attire vers un centre fixe,’’ C. R. 77, 849–853 (1873).
M. G. Darboux, ‘‘Recherche de la loi que dois suivre une force centrale pour que la trajectoire quellle determine soit toujour une conique,’’ C. R. 84, 760–762 (1877).
V. G. Imshenetsky, ‘‘Determination of the force moving a material point along a conic section in a function of its coordinates,’’ Soobshch. Khark. Mat. Ob-va, No. 1, 5–15 (1879).
Funding
This work was supported by the Russian Science Foundation and Moscow city no. 23-21-10065, https://rscf.ru/en/project/23-21-10065/.
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Mukharlyamov, R.G., Kaspirovich, I.E. Analysis of Possible Solutions of Some Inverse Dynamical Problem with Regard for Constraint Stabilization. Lobachevskii J Math 45, 472–477 (2024). https://doi.org/10.1134/S1995080224010384
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DOI: https://doi.org/10.1134/S1995080224010384