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

$d{\mathrm{F}\mathrm{l}}_X^t:TM\to TM$

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

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