Differentiation along the flow of a dynamical system

An operator defined as follows. Let

$$ \tag{* } \dot{x} = f ( x) $$

be an autonomous system, let $ x \in \mathbf R ^ {n} $, let $ f = ( f _ {1} \dots f _ {n} ) $, and let $ f _ {j} : G \rightarrow \mathbf R $ be smooth mappings, where $ G $ is a domain in $ \mathbf R ^ {n} $. Let a smooth mapping $ \phi : G \rightarrow \mathbf R $ be given. The derivative $ \theta _ {f} \phi $ along the flow of the system (*) of $ \phi $ at a point $ x ^ {0} \in G $ is defined by

$$ \left . ( \theta _ {f} \phi ) x ^ {0} = \sum _ {j = 1 } ^ { n } \frac{\partial \phi ( x ^ {0} ) }{\partial x _ {j} } f _ {j} ( x ^ {0} ) = \frac{d}{dt} ( \phi ( x ( t , x ^ {0} ) ) ) \right | _ {t = t ^ {0} } , $$

where $ x ( t , x ^ {0} ) $ is a solution of the system (*) such that $ x ( t ^ {0} , x ^ {0} ) = x ^ {0} $. The operator $ \theta _ {f} $ displays the following properties: 1) linearity in $ \phi $; and 2) $ \theta _ {f} ( \phi _ {1} \phi _ {2} ) = \phi _ {1} \theta _ {f} \phi _ {2} + \phi _ {2} \theta _ {f} \phi _ {1} $. The function $ ( \theta _ {f} \phi ) ( x) $ coincides with the derivative of $ \phi $ with respect to the vector field $ f $.

References

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

Comments

In terms of the canonical basis $ ( \partial / {\partial x _ {1} } \dots \partial / {\partial x _ {n} } ) $ of the tangent space $ T _ {x} \mathbf R ^ {n} $ at a point $ x $, the vector field $ f $ is written

$$ \sum f _ {j} ( x) \frac \partial {\partial x _ {j} } . $$

This first-order differential operator defines a derivation of the ring (cf. Derivation in a ring) of smooth functions $ C ^ \infty ( \mathbf R ^ {n} ) $ into itself. Moreover, this sets up a bijective correspondence between vector fields on $ \mathbf R ^ {n} $ and derivations on $ C ^ \infty ( \mathbf R ^ {n} ) $. 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 $ M $ as derivations of $ C ^ \infty ( M) $ and to observe subsequently that the notion corresponds to a section of the tangent bundle. In this setting a tangent vector at $ x \in M $ can be defined as a derivation on the local algebra of germs of smooth functions at $ x $ on $ M $. Thus, differentiation along the flow of a dynamical system given by the vector field $ f $ simply means applying the derivation on $ C ^ \infty ( M) $ given by $ f $.

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)