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 $\Omega \subset \mathbb \times M$ on $TM$

:$\begin X: \Omega \subset \mathbb \times M &\longrightarrow TM \\ (t,x) &\longmapsto X(t,x) = X_t(x) \in T_xM \end$

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

For every $t \in \mathbb$ such that the set

:$\Omega_t=\ \subset M$

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:

:$\frac=X(t,x)$

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

:$\alpha : I \subset \mathbb \longrightarrow M$

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

:$\frac \left.\_ =X(t_0,\alpha (t_0))$.

Equivalence with time-independent vector fields

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

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

:$\mathbb \times M \ni (t,p) \mapsto \dfrac\Biggl, _t + X(p) \in T_(\mathbb \times M)$

on $\mathbb \times M.$ In coordinates,

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

The system of autonomous differential equations for $\tilde$ 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 $\tilde,$ respectively.

Flow

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

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

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 $F_$ as $F_(p)=F(t,s,p)$

#If $(t_1,t_0,p) \in D(X)$ and $(t_2,t_1,F_(p)) \in D(X)$ then $F_ \circ F_(p)=F_(p)$
#$\forall t,s$, $F_$ is a diffeomorphism with inverse $F_$.

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 .\_ (F^*_ Y_t)_p = \left( F^*_ \left( _,Y_+ \frac \left .\_ 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:

:$\frac \left .\_ (F^*_ \eta_t)_p = \left( F^*_ \left( \mathcal_\eta_ + \frac \left .\_ \eta_t \right) \right)_p$

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.

Differential geometry
Vector calculus