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| In the [[mathematics|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.
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| Specifically, let ''X'' be a vector field on ''M''. Then ''X'' generates a [[one-parameter group]] of [[local diffeomorphism]]s ''Fl''<sub>X</sub><sup>t</sub>, the [[vector flow|flow]] along ''X''. The [[pushforward (differential)|differential]] of ''Fl''<sub>X</sub><sup>t</sup> gives, for each ''t'', a mapping
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| :<math> d\mathrm{Fl}_X^t : TM \to TM </math>
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| 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''<sub>X</sub><sup>t</sup>.
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| ==References==
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| * {{cite book|author=Shlomo Sternberg|authorlink=Shlomo Sternberg|title=Lectures on differential geometry|publisher=Prentice-Hall|year=1964|pages=p. 96}}
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| [[Category:Calculus of variations]]
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| {{geometry-stub}}
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Latest revision as of 03:58, 6 December 2014
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