# 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.