Time dependent vector field

A time dependent vector field on a manifold M is a map from an open subset $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}\; \backslash times\; M$ on $TM$

:$\backslash begin\{align\}$

X: \Omega \subset \mathbb{R} \times M &\longrightarrow TM \\

(t,x) &\longmapsto X(t,x) = X_t(x) \in T_xM

\end{align}

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

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

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

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

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

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

which is called nonautonomous by definition.

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

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

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

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

A time dependent vector field $X$ on $M$ can be thought of as a vector field $\backslash tilde\{X\}$ on $\backslash mathbb\{R\}\; \backslash times\; M,$ where $\backslash tilde\{X\}(t,p)\; \backslash in\; T\_\{(t,p)\}(\backslash mathbb\{R\}\; \backslash times\; M)$ does not depend on $t.$

Conversely, associated with a time-dependent vector field $X$ on $M$ is a time-independent one $\backslash tilde\{X\}$

:$\backslash mathbb\{R\}\; \backslash times\; M\; \backslash ni\; (t,p)\; \backslash mapsto\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; t\}\backslash Biggl|\_t\; +\; X(p)\; \backslash in\; T\_\{(t,p)\}(\backslash mathbb\{R\}\; \backslash times\; M)$

on $\backslash mathbb\{R\}\; \backslash times\; M.$ In coordinates,

:$\backslash tilde\{X\}(t,x)=(1,X(t,x)).$

The system of autonomous differential equations for $\backslash tilde\{X\}$ is equivalent to that of non-autonomous ones for $X,$ and $x\_t\; \backslash leftrightarrow\; (t,x\_t)$ is a bijection between the sets of integral curves of $X$ and $\backslash tilde\{X\},$ respectively.

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

:$F:D(X)\; \backslash subset\; \backslash mathbb\{R\}\; \backslash times\; \backslash Omega\; \backslash longrightarrow\; M$

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

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

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

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

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

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

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

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

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

This last identity is useful to prove the Darboux theorem.

• Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) {{isbn|0-387-95495-3}}. Graduate-level textbook on smooth manifolds.

Category:Differential geometry

Category:Vector calculus