Variation vector field

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Definition

Let M be a differential manifold.

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.