In the mathematical fields of the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

and

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, the variational vector field is a certain type of

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

defined on the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

of a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...

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 In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is in ...

local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Form ...

s ''Fl''_X^t, the flow along ''X''. The differential of ''Fl''_X^t gives, for each ''t'', a mapping :

d\mathrm_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

* Calculus of variations {{geometry-stub