Differentiation along the flow of a dynamical system

An operator defined as follows. Let

(*)

be an autonomous system, let , let , and let be smooth mappings, where is a domain in . Let a smooth mapping be given. The derivative along the flow of the system (*) of at a point is defined by

where is a solution of the system (*) such that . The operator displays the following properties: 1) linearity in ; and 2) . The function coincides with the derivative of with respect to the vector field .

References

[1]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)

Comments

In terms of the canonical basis of the tangent space at a point , the vector field is written

This first-order differential operator defines a derivation of the ring (cf. Derivation in a ring) of smooth functions into itself. Moreover, this sets up a bijective correspondence between vector fields on and derivations on . Using local coordinates this extends to the case of smooth differentiable manifolds. And indeed it is quite customary to define vector fields on a manifold as derivations of and to observe subsequently that the notion corresponds to a section of the tangent bundle. In this setting a tangent vector at can be defined as a derivation on the local algebra of germs of smooth functions at on . Thus, differentiation along the flow of a dynamical system given by the vector field simply means applying the derivation on given by .

References

[a1]	Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds and physics" , North-Holland (1977) pp. Sect. III.B (Translated from French)