About: Variational vector field

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 FlXt, the flow along X. The differential of FlXt gives, for each t, a mapping