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


 * $$\begin{align}

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

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

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


 * $$\Omega_t=\{x \in M \mid (t,x) \in \Omega \} \subset M$$

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:


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

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


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

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


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

Equivalence with time-independent vector fields
A time dependent vector field $$X$$ 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 $$\tilde{X}$$


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

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


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

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 $$F_{t,s}$$ 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) $$\forall t,s$$, $$F_{t,s}$$ 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:


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

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


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

This last identity is useful to prove the Darboux theorem.