Original manuscript: 2005/01/21
Results concerning C2-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 C2. 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 Mar 15 07:55:59 2024