# Time dependent vector field

(Redirected from 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 ${\displaystyle TM}$

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

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

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

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

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

## 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 ${\displaystyle {\mathbb {R}}\times M}$ even though its value on a point ${\displaystyle (t,x)}$ does not depend on the component ${\displaystyle t\in {\mathbb {R}}}$.

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

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

for each ${\displaystyle (t,x)\in \Omega }$, where we identify ${\displaystyle T_{(t,x)}({\mathbb {R}}\times M)}$ with ${\displaystyle {\mathbb {R}}\times T_{x}M}$. We can also write it as:

${\displaystyle {\tilde {X}}={\frac {\partial {}}{\partial {t}}}+X}$.

To each integral curve of X, we can associate one integral curve of ${\displaystyle {\tilde {X}}}$, and vice versa.

## 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.