In the mathematical fields of the calculus of variations and differential geometry, the variational vector field is a certain type of vector field defined on the tangent bundle of a differentiable manifold which gives rise to variations along a vector field in the manifold itself.
Specifically, let X be a vector field on M. Then X generates a one-parameter group of local diffeomorphisms Fl<sub>X</sub><sup>t</sup>, the flow along X. The differential of Fl<sub>X</sub><sup>t</sup> gives, for each t, a mapping
where TM denotes the tangent bundle of M. This is a one-parameter group of local diffeomorphisms of the tangent bundle. The variational vector field of X, denoted by T(X) is the tangent to the flow of d Fl<sub>X</sub><sup>t</sup>.