// Workers AI · dad joke modeWhy did the variational vector field go to therapy? It had variable mood.

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.

References

• Shlomo Sternberg (1964). Lectures on differential geometry. Prentice-Hall. pp. 96.