Geodesic
The Geometry of d 2 y 1/dt 2 = f (y, ẏ, t) and d 2 y 2/dt 2 = g(y, ẏ, t), and Euclidean Spaces
Canadian mathematical bulletin, Tome 49 (2006) no. 2, pp. 170-184

Voir la notice de l'article provenant de la source Cambridge University Press

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations ${{d}^{2}}{{y}^{1}}/d{{t}^{2}}=f\left( y,\dot{y},t \right)$ and ${{d}^{2}}{{y}^{2}}/d{{t}^{2}}=g\left( y,\dot{y},t \right)$ be reparameterized by $t\to T\left( y,t \right)$ so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.
DOI : 10.4153/CMB-2006-018-7
Mots-clés : 34A26 
Atkins, Richard. The Geometry of d 2 y 1/dt 2 = f (y, ẏ, t) and d 2 y 2/dt 2 = g(y, ẏ, t), and Euclidean Spaces. Canadian mathematical bulletin, Tome 49 (2006) no. 2, pp. 170-184. doi: 10.4153/CMB-2006-018-7
@article{10_4153_CMB_2006_018_7,
     author = {Atkins, Richard},
     title = {The {Geometry} of d 2 y 1/dt 2 = f (y, ẏ, t) and d 2 y 2/dt 2 = g(y, ẏ, t), and {Euclidean} {Spaces}},
     journal = {Canadian mathematical bulletin},
     pages = {170--184},
     year = {2006},
     volume = {49},
     number = {2},
     doi = {10.4153/CMB-2006-018-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-018-7/}
}
TY  - JOUR
AU  - Atkins, Richard
TI  - The Geometry of d 2 y 1/dt 2 = f (y, ẏ, t) and d 2 y 2/dt 2 = g(y, ẏ, t), and Euclidean Spaces
JO  - Canadian mathematical bulletin
PY  - 2006
SP  - 170
EP  - 184
VL  - 49
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-018-7/
DO  - 10.4153/CMB-2006-018-7
ID  - 10_4153_CMB_2006_018_7
ER  - 
%0 Journal Article
%A Atkins, Richard
%T The Geometry of d 2 y 1/dt 2 = f (y, ẏ, t) and d 2 y 2/dt 2 = g(y, ẏ, t), and Euclidean Spaces
%J Canadian mathematical bulletin
%D 2006
%P 170-184
%V 49
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-018-7/
%R 10.4153/CMB-2006-018-7
%F 10_4153_CMB_2006_018_7

[1] [1] Cartan, E., Observations sur le mémoire précédent. Math. Z. 37(1933), 619–622. Google Scholar

[2] [2] Chern, S. S., Sur la géometrie d’un système d’équations différentielles du second ordre. Bull. Sci. Math. 63(1939), 206–212. Google Scholar

[3] [3] Gardner, R. B., and Shadwick, W., Overdetermined equivalence problems with an application to feedback equivalence. Contemporary Mathematics 68, American Mathematical Society, Providence, RI, 1987, pp. 111–119. Google Scholar

[4] [4] Kosambi, D. D., Parallelism and path-spaces. Math. Z. 37(1933), 608–618. Google Scholar

Cité par Sources :