Variation vector field: Difference between revisions

Latest revision as of 20:13, 18 May 2008

Definition

Let $\alpha :(-\epsilon ,\epsilon )\times [0,1]\to M$ be a piecewise smooth variation of a curve $\omega :[0,1]\to M$ . The variation vector field of $\alpha$ is defined as the following element of the tangent space $T_{\omega }\Omega$ :

$W(t):={\frac {\partial \alpha (u,t)}{\partial u}}|_{u=0}$

Facts

Given any element of the tangent space of a piecewise smooth path in the path space, there exists a piecewise smooth variation thereof for which it is the variation vector field. This is obtained by simply exponentiating the vector field.

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 Let <math>\alpha: (-\epsilon,\epsilon) \times [0,1] \to M</math> be a [[piecewise smooth variation]] of a curve <math>\omega:[0,1] \to M</math>. The '''variation vector field''' of <math>\alpha</math> is defined as the following element of the tangent space <math>T_\omega\Omega</math>:
-<math>W(t) := \frac{\partual \alpha(u,t)}{\partial u}|_{u=0}</math>
+<math>W(t) := \frac{\partial \alpha(u,t)}{\partial u}|_{u=0}</math>
 ==Facts==
-Given any element of the tangent space of a [[piecewise smooth path]] in the [[path space]], there exists a [[piecewise smooth variation]] thereof for which it is the variation vector field. This is obtained by simply exponentiating the vector field. The variation obtained as the exponential of <math>\omega</math> is termed <math>E(\omega)</math>.
+Given any element of the tangent space of a [[piecewise smooth path]] in the [[path space]], there exists a [[piecewise smooth variation]] thereof for which it is the variation vector field. This is obtained by simply exponentiating the vector field.