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Search: MSC category 34A26 ( Geometric methods in differential equations )

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1. CMB 2006 (vol 49 pp. 170)

Atkins, Richard
The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces
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 $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and $d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by $t\rightarrow T(y,t)$ 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.


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