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) |
Differentiation along the flow of a dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differentiation_along_the_flow_of_a_dynamical_system&oldid=18037