Differential geometry of surfaces: Difference between revisions

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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.
 
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
 
:<math> d\mathrm{Fl}_X^t : TM \to TM </math>
 
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>.
 
==References==
 
* {{cite book|author=Shlomo Sternberg|authorlink=Shlomo Sternberg|title=Lectures on differential geometry|publisher=Prentice-Hall|year=1964|pages=p. 96}}
 
[[Category:Calculus of variations]]
{{geometry-stub}}

Revision as of 21:21, 7 December 2013

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 FlXt, the flow along X. The differential of FlXt gives, for each t, a mapping

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 FlXt.

References

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