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

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

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

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

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

is nonempty, ${\displaystyle X_{t}}$ is a vector field in the usual sense defined on the open set ${\displaystyle \Omega _{t}\subset M}$.

Associated differential equation

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

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

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

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

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

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

Equivalence with time-independent vector fields

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

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

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

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

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

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

Flow

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

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

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

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

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

Properties

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

1. If ${\displaystyle (t_{1},t_{0},p)\in D(X)}$ and ${\displaystyle (t_{2},t_{1},F_{t_{1},t_{0}}(p))\in D(X)}$ then ${\displaystyle F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)}$
2. ${\displaystyle \forall t,s}$, ${\displaystyle F_{t,s}}$ is a diffeomorphism with inverse ${\displaystyle F_{s,t}}$.

Applications

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

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

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

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

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.