Brachistochrone on the sphere

Ferreira, Rafael Xavier Deiga; Salazar, Hector Leny Carrion

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Revista Brasileira de Ensino de Física

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Sumário

Artigos Gerais • Rev. Bras. Ensino Fís. 41 (4) • 2019 • https://doi.org/10.1590/1806-9126-RBEF-2019-0020 copy

Brachistochrone on the sphere

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The problem is to find the path (between two given points) of minimum time of a body, which is released from rest, restricted to a sphere surface and under a constant gravitational field. Using Euler-Lagrange equation, it is found a expression to the total travel time. The problem was solved numerically using computation physics techniques and some numerical values were obtain using the software Mathematica, which were compared with the free fall time and with the time that the body would do if it moved in the sphere geodesic (which connects the two given points). Besides, it was studied the time dependence of the body with the position in which it is released in a same path of minimum time (analogous to the tautochrone problem in the plane).

Keywords:
Euler-Lagrange equation; Brachistochrone; Shooting method

$ϕ_{0}$	$θ_{0}$	$ϕ_{f}$	$θ_{f}$	QL	Mín	Geod
$\frac{π}{3}$	$\frac{π}{6}$	$\frac{π}{10}$	$\frac{2 π}{5}$	0,334	0,594	0,677
$\frac{π}{3}$	$\frac{π}{6}$	$\frac{π}{12}$	$\frac{5 π}{12}$	0,348	0,615	0,708
$\frac{π}{3}$	$\frac{π}{4}$	$\frac{π}{4}$	$\frac{3 π}{10}$	0,154	0,297	0,677
$\frac{π}{4}$	$\frac{π}{4}$	$\frac{π}{10}$	$\frac{2 π}{5}$	0,282	0,433	0,482
$\frac{π}{5}$	$\frac{π}{3}$	$\frac{π}{7}$	$\frac{7 π}{18}$	0,178	0,251	0,276
$\frac{π}{6}$	$\frac{π}{6}$	$\frac{π}{15}$	$\frac{3 π}{10}$	0,236	0,423	0,443
$\frac{π}{12}$	$\frac{π}{6}$	0	$\frac{π}{2}$	0,416	0,593	0,599
$\frac{π}{13}$	$\frac{π}{2}$	0	$\frac{5 π}{6}$	0,416	0,471	0,473
$\frac{π}{14}$	$\frac{π}{2}$	$\frac{π}{20}$	$\frac{7 π}{10}$	0,343	0,358	0,359
0	$\frac{π}{6}$	- $\frac{π}{3}$	$\frac{3 π}{5}$	0,485	0,747	0,856
0	$\frac{π}{6}$	- $\frac{π}{4}$	$\frac{11 π}{18}$	0,492	0,714	0,765
0	$\frac{π}{6}$	- $\frac{π}{5}$	$\frac{5 π}{8}$	0,500	0,703	0,733
- $\frac{π}{5}$	$\frac{3 π}{5}$	- $\frac{π}{4}$	$\frac{3 π}{4}$	0,282	0,329	0,331
- $\frac{π}{4}$	$\frac{3 π}{4}$	- $\frac{π}{3}$	$\frac{5 π}{6}$	0,178	0,311	0,319

$ϕ_{0}$	$θ_{0}$	$ϕ_{f}$	$θ_{f}$	QL	Mín	Geod
$\frac{π}{3}$	$7, {5.10}^{- 2}$	$\frac{π}{4}$	$\frac{5 π}{6}$	0,610	1,396	1,403
$\frac{π}{3}$	$7, {9.10}^{- 4}$	$\frac{π}{4}$	$\frac{5 π}{6}$	0,611	2,836	2,846
$\frac{π}{3}$	$7, {9.10}^{- 6}$	$\frac{π}{4}$	$\frac{5 π}{6}$	0,611	4,292	4,301
$\frac{π}{3}$	$\frac{π}{6}$	$\frac{π}{4}$	$3, 07$	0,610	0,863	0,864
$\frac{π}{3}$	$\frac{π}{6}$	$\frac{π}{4}$	$3, 1408$	0,611	0,875	0,875
$\frac{π}{3}$	$\frac{π}{6}$ & $\frac{π}{4}$	$3, 1415$	0,611	0,875	0,875
$\frac{π}{3}$	$7, {5.10}^{- 2}$	$\frac{π}{4}$	$3, 07$	0,632	1,467	1,469
$\frac{π}{3}$	$7, {9.10}^{- 4}$	$\frac{π}{4}$	$3, 1408$	0,632	2,918	2,921
$\frac{π}{3}$	$7, {9.10}^{- 6}$	$\frac{π}{4}$	$3, 1415$	0,632	4,385	Erro

$θ$ (rad)	Tempo (s)
0,524	0,875
0,733	0,778
0,942	0,710
1,152	0,658
1,361	0,618
1,571	0,586
1,780	0,561
1,990	0,541
2,199	0,526
2,409	0,514
2,618	0,505

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E-mail: marcio@sbfisica.org.br

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[1] ^*Endereço de correspondência: rafaeldeiga@hotmail.com