Geodesic
On the trajectories of bodies in non-inertial reference frames
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 84 (2023), pp. 68-80 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper considers a trajectory of the body moving under the influence of the force $\mathbf{F}$ in a non-inertial reference frame (NRF), which is "tied" to a given curve $y=y(x)$ and is described by a natural movable basis $\tau$-$\mathbf{n}$. For this NRF, a system of linear differential equations is obtained to simulate various types of trajectories resulting from the action of certain forces. The common Cartesian coordinate system is chosen as a fixed basis $\mathbf{i}$-$\mathbf{j}$. Several examples of motion along the given trajectories $y=y(x)$ are considered with gravity as an acting force $\mathbf{F}$. For these specific cases, the analytic expressions for absolute (in the system $\mathbf{i}$-$\mathbf{j}$), relative (in the system $\tau$-$\mathbf{n}$), and translational accelerations are given. The corresponding trajectories of motion under free fall conditions in terms of NRF are constructed. The following trajectories $y=y(x)$ are studied: uneven motion along a straight line, a brachistochrone, and a circle. Using computer modeling tools, the results are presented as plots showing the qualitative difference between the trajectories of the same body in the inertial and non-inertial frames of reference. The considered limiting cases of motion confirm the validity of the obtained general system of equations in the NRF.
Keywords: movable basis, absolute and relative motion, non-inertial reference frame. 
@article{VTGU_2023_84_a5,
     author = {S. B. Bogdanova and S. O. Gladkov},
     title = {On the trajectories of bodies in non-inertial reference frames},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {68--80},
     year = {2023},
     number = {84},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2023_84_a5/}
}
TY  - JOUR
AU  - S. B. Bogdanova
AU  - S. O. Gladkov
TI  - On the trajectories of bodies in non-inertial reference frames
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2023
SP  - 68
EP  - 80
IS  - 84
UR  - http://geodesic.mathdoc.fr/item/VTGU_2023_84_a5/
LA  - ru
ID  - VTGU_2023_84_a5
ER  - 
%0 Journal Article
%A S. B. Bogdanova
%A S. O. Gladkov
%T On the trajectories of bodies in non-inertial reference frames
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2023
%P 68-80
%N 84
%U http://geodesic.mathdoc.fr/item/VTGU_2023_84_a5/
%G ru
%F VTGU_2023_84_a5
S. B. Bogdanova; S. O. Gladkov. On the trajectories of bodies in non-inertial reference frames. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 84 (2023), pp. 68-80. http://geodesic.mathdoc.fr/item/VTGU_2023_84_a5/

[1] L. D. Landau, Kurs teoreticheskoi fiziki, v 10 t., v. 1, Mekhanika, Fizmatlit, M., 1988

[2] D. V. Sivukhin, Obschii kurs fiziki, v 5 t., v. 1, Mekhanika, Fizmatlit, M., 2010

[3] S. P. Strelkov, Mekhanika, Fizmatlit, M., 1975

[4] A. I. Ivanov, “O brakhistokhrone chastitsy peremennoi massy s postoyannym otnosheniem kolichestva prisoedinyaemykh i otdelyaemykh chastits”, Doklady AN USSR. Seriya A, 1968, 683–686 | Zbl

[5] G. Dupeux, A. Le Goff, D. Quere, C. Clanet, “The spinning ball spiral”, New Journal of Physics, 12 (2010), 093004 | DOI

[6] H. H. Denman, “Remarks on brachistochrone tautochrone problems”, Amer. J. Phys, 53 (1985), 224–227 | DOI

[7] T. V. Bordovitsyna, A. G. Aleksandrova, I. N. Chuvashov, “Chislennoe modelirovanie dinamiki okolozemnykh kosmicheskikh ob'ektov iskusstvennogo proiskhozhdeniya s ispolzovaniem parallelnykh vychislenii”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2011, no. 4 (16), 34–48

[8] O. V. Matvienko, A. O. Andropova, A. V. Andiasyan, N. A. Mamadraimova, “Matematicheskoe modelirovanie dvizheniya sfericheskoi chastitsy po naklonnoi poverkhnosti v sdvigovom potoke”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 52, 75–88 | DOI | Zbl

[9] N. Ashby, W. E. Brittin, W. F. Love, W. Wyss, “Brachistochrone with Coulomb friction”, Amer. J. Phys., 43:10 (1975), 902–905 | DOI | MR

[10] J. C. Hayen, “Brachistochrone with Coulomb friction”, Int. J. Non-Linear Mech, 40:8 (2005), 1057–1075 | DOI | MR | Zbl

[11] S. O. Gladkov, S. B. Bogdanova, “Geometricheskii fazovyi perekhod v zadache o brakhistokhrone”, Uchenye zapiski fizicheskogo fakulteta MGU, 2016, no. 1, 161101, 1–5

[12] S. O. Gladkov, S. B. Bogdanova, “O forme brakhistokhrony, vraschayuscheisya v vertikalnoi ploskosti”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2022, no. 78, 86–98

[13] S. O. Gladkov, S. B. Bogdanova, “Analytical and numerical solution of the problem on brachistochrones in some general cases”, Journal of Mathematical Sciences, 245:4 (2020), 528–537 | DOI | Zbl

[14] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, v. 1, Fizmatlit, M., 1962

[15] V. I. Smirnov, Kurs vysshei matematiki, v. 3, Fizmatlit, M., 1967

[16] S. B. Bogdanova, N. V. Grigorevskii, “Ob odnom metodicheskom podkhode k opisaniyu svoistv tsikloidy”, Fizicheskoe obrazovanie v vuzakh, 27:3 (2021), 47–56