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"_Xt gives, for each "t", a mapping}

:$dmathrm\{Fl\}\_X^t\; :\; TM\; o\; TM$

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.

References

*cite book|author=Shlomo Sternberg|authorlink=Shlomo Sternberg|title=Lectures on differential geometry|publisher=Prentice-Hall|year=1964|pages=p. 96