**Original manuscript:** 2005/01/21

Results concerning *C*^{2}-minimizing curves on manifolds are
presented. A coordinate-free derivation of the Euler-Lagrange equation is
presented. Using a variational approach, two vector fields are defined along
the minimizing curve; the tangent to the curve, and the infinitesimal
variation. The derivation presented involves complete lifts of arbitrary
extensions of these vector fields and it is shown that the derivation is
independent of the particular choice of extensions. Special care is also
taken to ensure that the derivation does not require any additional
differentiability constraints, other than the curve being of class
*C*^{2}.
Minimizing curves are also characterized in terms of their local behaviour.
It is shown that if a curve is minimizing then any sub-arc of the curve is
also minimizing. An important corollary of this result is that a curve, on a
manifold will be minimizing only if any collection of admissible charts which
cover the curve have minimizing local representations.

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Last Updated: Fri Jul 10 07:29:14 2020