# Time dependent vector field

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In [mathematics](/source/Mathematics), a **time dependent vector field** is a construction in [vector calculus](/source/Vector_calculus) which generalizes the concept of [vector fields](/source/Vector_field). It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a [vector](/source/Vector_(geometric)) to every point in a [Euclidean space](/source/Euclidean_space) or in a [manifold](/source/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](/source/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](/source/Differential_equation):

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

which is called [nonautonomous](/source/Autonomous_system_(mathematics)) by definition.

## Integral curve

An [integral curve](/source/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](/source/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](/source/Flow_(mathematics)) 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)
1. \forall t,s, F_{t,s} is a [diffeomorphism](/source/Diffeomorphism) with [inverse](/source/Inverse_function) 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](/source/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.

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Adapted from the Wikipedia article [Time dependent vector field](https://en.wikipedia.org/wiki/Time_dependent_vector_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Time_dependent_vector_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.