Time dependent vector field

In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.

Definition

A time dependent vector field on a manifold "M" is a map from an open subset $Omega\; subset\; Bbb\{R\}\; imes\; M$ on $TM$

:$X:\; Omega\; subset\; Bbb\{R\}\; imes\; M\; longrightarrow\; TM$

::::$(t,x)\; longmapsto\; X(t,x)=X\_t(x)\; in\; T\_xM$

such that for every $(t,x)\; in\; Omega$, $X\_t(x)$ is an element of $T\_xM$.

For every $t\; in\; Bbb\{R\}$ such that the set

:$Omega\_t=\{x\; in\; M\; |\; (t,x)\; in\; Omega\; \}\; subset\; M$

is nonempty, $X\_t$ is a vector field in the usual sense defined on the open set $Omega\_t\; subset\; M$.

Associated differential equation

Given a time dependent vector field "X" on a manifold "M", we can associate to it the following differential equation:

:$frac\{dx\}\{dt\}=X(t,x)$

which is called nonautonomous by definition.

Integral curve

An integral curve of the equation above (also called an integral curve of "X") is a map

:$alpha\; :\; I\; subset\; Bbb\{R\}\; longrightarrow\; M$

such that $forall\; t\_0\; in\; I$, $(t\_0,alpha\; (t\_0))$ is an element of the domain of definition of "X" and

:$frac\{d\; alpha\}\{dt\}\; left.\{!!frac\{\}\{\; ight|\_\{t=t\_0\}\; =X(t\_0,alpha\; (t\_0))$.

Relationship with vector fields in the usual sense

A vector field in the usual sense can be thought of as a time dependent vector field defined on $Bbb\{R\}\; imes\; M$ even though its value on a point $(t,x)$ does not depend on the component $t\; in\; Bbb\{R\}$.

Conversely, given a time dependent vector field "X" defined on $Omega\; subset\; Bbb\{R\}\; imes\; M$, we can associate to it a vector field in the usual sense $ilde\{X\}$ on $Omega$ such that the autonomous differential equation associated to $ilde\{X\}$ is essentially equivalent to the nonautonomous differential equation associated to "X". It suffices to impose:

:$ilde\{X\}(t,x)=(1,X(t,x))$

for each $(t,x)\; in\; Omega$, where we identify $T\_\{(t,x)\}(Bbb\{R\}\; imes\; M)$ with $Bbb\{R\}\; imes\; T\_x\; M$. We can also write it as:

:$ilde\{X\}=frac\{partial\{\{partial\{t+X$.

To each integral curve of "X", we can associate one integral curve of $ilde\{X\}$, and viceversa.

Flow

The flow of a time dependent vector field "X", is the unique differentiable map

:$F:D(X)\; subset\; Bbb\{R\}\; imes\; Omega\; longrightarrow\; M$

such that for every $(t\_0,x)\; in\; Omega$,

:$t\; longrightarrow\; F(t,t\_0,x)$

is the integral curve of "X" $alpha$ that verifies $alpha\; (t\_0)\; =\; x$.

Properties

We define $F\_\{t,s\}$ as $F\_\{t,s\}(p)=F(t,s,p)$

#If $(t\_1,t\_0,p)\; in\; D(X)$ and $(t\_2,t\_1,F\_\{t\_1,t\_0\}(p))\; in\; D(X)$ then $F\_\{t\_2,t\_1\}\; circ\; F\_\{t\_1,t\_0\}(p)=F\_\{t\_2,t\_0\}(p)$
#$forall\; t,s$, $F\_\{t,s\}$ is a diffeomorphism with inverse $F\_\{s,t\}$.

Applications

Let "X" and "Y" be smooth time dependent vector fields and $F$ the flow of "X". The following identity can be proved:

:$frac\{d\}\{dt\}\; left\; .\{!!frac\{\}\{\; ight|\_\{t=t\_1\}\; (F^*\_\{t,t\_0\}\; Y\_t)\_p\; =\; left(\; F^*\_\{t\_1,t\_0\}\; left(\; [X\_\{t\_1\},Y\_\{t\_1\}]\; +\; frac\{d\}\{dt\}\; left\; .\{!!frac\{\}\{\; ight|\_\{t=t\_1\}\; Y\_t\; ight)\; ight)\_p$

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that $eta$ is a smooth time dependent tensor field:

:$frac\{d\}\{dt\}\; left\; .\{!!frac\{\}\{\; ight|\_\{t=t\_1\}\; (F^*\_\{t,t\_0\}\; eta\_t)\_p\; =\; left(\; F^*\_\{t\_1,t\_0\}\; left(\; mathcal\{L\}\_\{X\_\{t\_1eta\_\{t\_1\}\; +\; frac\{d\}\{dt\}\; left\; .\{!!frac\{\}\{\; ight|\_\{t=t\_1\}\; eta\_t\; ight)\; ight)\_p$

This last identity is useful to prove the Darboux theorem.

References

* Lee, John M., "Introduction to Topological Manifolds", Springer-Verlag, New York (2000), ISBN 0-387-98759-2. "Introduction to Smooth Manifolds", Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.