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 $\Omega \subset \mathbb {R} \times M$ on $TM$

${\begin{aligned}X:\Omega \subset \mathbb {R} \times M&\longrightarrow TM\\(t,x)&\longmapsto X(t,x)=X_{t}(x)\in T_{x}M\end{aligned}}$

such that for every $(t,x)\in \Omega$, $X_{t}(x)$ is an element of $T_{x}M$.

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

$\Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M$

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

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

such that $\forall t_{0}\in I$, $(t_{0},\alpha (t_{0}))$ is an element of the domain of definition of X and

${\frac {d\alpha }{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{0}}=X(t_{0},\alpha (t_{0}))$.

Equivalence with time-independent vector fields

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

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

$\mathbb {R} \times M\ni (t,p)\mapsto {\dfrac {\partial }{\partial t}}{\Biggl |}_{t}+X(p)\in T_{(t,p)}(\mathbb {R} \times M)$

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

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

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

Flow

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

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

such that for every $(t_{0},x)\in \Omega$,

$t\longrightarrow F(t,t_{0},x)$

is the integral curve $\alpha$ of X that satisfies $\alpha (t_{0})=x$.

Properties

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

1. If $(t_{1},t_{0},p)\in D(X)$ and $(t_{2},t_{1},F_{t_{1},t_{0}}(p))\in D(X)$ then $F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)$
2. $\forall t,s$, $F_{t,s}$ 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:

${\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}(F_{t,t_{0}}^{*}Y_{t})_{p}=\left(F_{t_{1},t_{0}}^{*}\left([X_{t_{1}},Y_{t_{1}}]+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}Y_{t}\right)\right)_{p}$

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:

${\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}(F_{t,t_{0}}^{*}\eta _{t})_{p}=\left(F_{t_{1},t_{0}}^{*}\left({\mathcal {L}}_{X_{t_{1}}}\eta _{t_{1}}+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{t=t_{1}}\eta _{t}\right)\right)_{p}$

This last identity is useful to prove the Darboux theorem.

References

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