Geodesic
On a class of planar geometrical curves with constant reaction forces acting to particles moving along them
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences ICMMAS-17, St. Petersburg Polytechnic University, July 24-28, 2017, Tome 160 (2019), pp. 28-31.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we find the dependence of the reaction force $N(y)$ of a curved trough of arbitrary shape described by a function $y(x)$. Based on the extremum condition ${dN}/{dx}$ valid for any point of the abscissa axis, we examine the equation $N(y,y',y'')=\operatorname{const}$ whose solutions determine the desired class of curves $y(x)$. We obtain an analytic solution of this equations and perform numerical simulations for various values of parameters. Examples of functions $y(x)$ for which $N=\operatorname{const}$ are presented.
Keywords: reaction force, curved trough, nonlinear differential equation, numerical solution. 
@article{INTO_2019_160_a3,
     author = {S. O. Gladkov and S. B. Bogdanova},
     title = {On a class of planar geometrical curves with constant reaction forces acting to particles moving along them},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {28--31},
     publisher = {mathdoc},
     volume = {160},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_160_a3/}
}
TY  - JOUR
AU  - S. O. Gladkov
AU  - S. B. Bogdanova
TI  - On a class of planar geometrical curves with constant reaction forces acting to particles moving along them
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 28
EP  - 31
VL  - 160
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_160_a3/
LA  - ru
ID  - INTO_2019_160_a3
ER  - 
%0 Journal Article
%A S. O. Gladkov
%A S. B. Bogdanova
%T On a class of planar geometrical curves with constant reaction forces acting to particles moving along them
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 28-31
%V 160
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_160_a3/
%G ru
%F INTO_2019_160_a3
S. O. Gladkov; S. B. Bogdanova. On a class of planar geometrical curves with constant reaction forces acting to particles moving along them. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences ICMMAS-17, St. Petersburg Polytechnic University, July 24-28, 2017, Tome 160 (2019), pp. 28-31. http://geodesic.mathdoc.fr/item/INTO_2019_160_a3/

[1] Gladkov S. O., Sbornik zadach po teoreticheskoi i matematicheskoi fizike, Fizmatlit, M., 2010

[2] Gladkov S. O., “O traektorii dvizheniya tela, vkhodyaschego v zhidkost pod proizvolnym uglom”, Uch. zap. fiz. f-ta MGU, 2016, no. 4, 164002-1-5

[3] Gladkov S. O., Bogdanova S. B., “Geometricheskii fazovyi perekhod v zadache o brakhistokhrone”, Uch. zap. fiz. f-ta MGU, 2016, no. 1, 161101-1-5

[4] Gladkov S. O., Bogdanova S. B., “Obobschennye dinamicheskie uravneniya ploskogo krivolineinogo dvizheniya materialnogo tela po zhelobu s uchetom sil treniya, i ikh chislennyi analiz v nekotorykh chastnykh sluchayakh”, Uch. zap. fiz. f-ta MGU, 2017, no. 1, 171101-1-5

[5] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 2, Nauka, M., 1965

[6] Elsgolts L. E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969