Definitions

# 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 $Omega subset Bbb\left\{R\right\} times M$ on $TM$

$X: Omega subset Bbb\left\{R\right\} times M longrightarrow TM$

$\left(t,x\right) longmapsto X\left(t,x\right)=X_t\left(x\right) in T_xM$

such that for every $\left(t,x\right) in Omega$, $X_t\left(x\right)$ is an element of $T_xM$.

For every $t in Bbb\left\{R\right\}$ such that the set

$Omega_t=\left\{x in M | \left(t,x\right) in Omega \right\} subset M$

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:

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

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

$alpha : I subset Bbb\left\{R\right\} longrightarrow M$

such that $forall t_0 in I$, $\left(t_0,alpha \left(t_0\right)\right)$ is an element of the domain of definition of X and

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

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

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

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

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

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

To each integral curve of X, we can associate one integral curve of $tilde\left\{X\right\}$, and viceversa.

## Flow

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

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

such that for every $\left(t_0,x\right) in Omega$,

$t longrightarrow F\left(t,t_0,x\right)$

is the integral curve of X $alpha$ that verifies $alpha \left(t_0\right) = x$.

### Properties

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

1. If $\left(t_1,t_0,p\right) in D\left(X\right)$ and $\left(t_2,t_1,F_\left\{t_1,t_0\right\}\left(p\right)\right) in D\left(X\right)$ then $F_\left\{t_2,t_1\right\} circ F_\left\{t_1,t_0\right\}\left(p\right)=F_\left\{t_2,t_0\right\}\left(p\right)$
2. $forall t,s$, $F_\left\{t,s\right\}$ is a diffeomorphism with inverse $F_\left\{s,t\right\}$.

## Applications

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

$frac\left\{d\right\}\left\{dt\right\} left .\left\{!!frac\left\{\right\}\left\{\right\}\right\}right|_\left\{t=t_1\right\} \left(F^*_\left\{t,t_0\right\} Y_t\right)_p = left\left(F^*_\left\{t_1,t_0\right\} left\left(\left[X_\left\{t_1\right\},Y_\left\{t_1\right\}\right] + frac\left\{d\right\}\left\{dt\right\} left .\left\{!!frac\left\{\right\}\left\{\right\}\right\}right|_\left\{t=t_1\right\} Y_t right\right) right\right)_p$

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:

$frac\left\{d\right\}\left\{dt\right\} left .\left\{!!frac\left\{\right\}\left\{\right\}\right\}right|_\left\{t=t_1\right\} \left(F^*_\left\{t,t_0\right\} eta_t\right)_p = left\left(F^*_\left\{t_1,t_0\right\} left\left(mathcal\left\{L\right\}_\left\{X_\left\{t_1\right\}\right\}eta_\left\{t_1\right\} + frac\left\{d\right\}\left\{dt\right\} left .\left\{!!frac\left\{\right\}\left\{\right\}\right\}right|_\left\{t=t_1\right\} eta_t right\right) right\right)_p$

This last identity is useful to prove the Darboux theorem.

## References

• Lee, John M., Introduction to Topological Manifolds, Springer-Verlag, New York (2000), ISBN 0-387-98759-2. Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
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