Time dependent vector field

Short description: Vector calculus construction

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

$\begin{array}{rl}X:\mathrm{\Omega }\subset \mathbb{ℝ}\times M& ⟶TM\\ (t,x)& ⟼X(t,x)=X_t(x)\in T_{x}M\end{array}$

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

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

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

is nonempty, $X_t$ is a vector field in the usual sense defined on the open set ${\mathrm{\Omega }}_t\subset M$.

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

$\alpha :I\subset \mathbb{ℝ}⟶M$

such that $\mathrm{\forall }{t}_0\in I$, $(t_0,\alpha (t_0))$ is an element of the domain of definition of X and

$\frac{d\alpha }{dt}{\frac{}{}|}_{t=t_0}=X(t_0,\alpha (t_0))$.

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 $\tilde{X}$

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

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

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

such that for every $(t_0,x)\in \mathrm{\Omega }$,

$t⟶F(t,t_0,x)$

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. $\mathrm{\forall }t,s$, $F_{t,s}$ 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:

$\frac{d}{dt}{\frac{}{}|}_{t=t_1}(F_{t,t_0}^{*}{Y}_t{)}_p={\left(F_{t_1,t_0}^{*}([X_{t_1},Y_{t_1}]+\frac{d}{dt}{\frac{}{}|}_{t=t_1}{Y}_t)\right)}_p$

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:

$\frac{d}{dt}{\frac{}{}|}_{t=t_1}(F_{t,t_0}^{*}{\eta }_t{)}_p={\left(F_{t_1,t_0}^{*}({{ℒ}}_{X_{t_1}}{\eta }_{t_1}+\frac{d}{dt}{\frac{}{}|}_{t=t_1}{\eta }_t)\right)}_p$

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.