Variational vector field
# 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 $Fl_X^t,$ the flow along X. The differential of $Fl_X^t$ gives, for each t, a mapping

    $d\mathrm {Fl} _{X}^{t}:TM\to TM$

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 $Fl_X^t.$

