View source for Differentiation along the flow of a dynamical system

An operator defined as follows. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

be an [[Autonomous system|autonomous system]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323102.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323103.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323104.png" /> be smooth mappings, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323105.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323106.png" />. Let a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323107.png" /> be given. The derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323108.png" /> along the flow of the system (*) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d0323109.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231010.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231012.png" /> is a solution of the system (*) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231013.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231014.png" /> displays the following properties: 1) linearity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231015.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231016.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231017.png" /> coincides with the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231018.png" /> with respect to the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231019.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>



====Comments====
In terms of the canonical basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231020.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231021.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231022.png" />, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231023.png" /> is written

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231024.png" /></td> </tr></table>

This first-order differential operator defines a derivation of the ring (cf. [[Derivation in a ring|Derivation in a ring]]) of smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231025.png" /> into itself. Moreover, this sets up a bijective correspondence between vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231026.png" /> and derivations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231027.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231028.png" /> as derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231029.png" /> and to observe subsequently that the notion corresponds to a section of the tangent bundle. In this setting a tangent vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231030.png" /> can be defined as a derivation on the local algebra of germs of smooth functions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231032.png" />. Thus, differentiation along the flow of a dynamical system given by the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231033.png" /> simply means applying the derivation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231034.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032310/d03231035.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Choquet-Bruhat,   C. DeWitt-Morette,   M. Dillard-Bleick,   "Analysis, manifolds and physics" , North-Holland  (1977)  pp. Sect. III.B  (Translated from French)</TD></TR></table>