Time dependent vector field

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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.

1 Definition
2 Associated differential equation
3 Integral curve
4 Relationship with vector fields in the usual sense
5 Flow
- 5.1 Properties
6 Applications
7 References

[edit] Definition

A time dependent vector field on a manifold M is a map from an open subset $\Omega \subset \Bbb{R} \times M$ on $T M$

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

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

such that for every $(t,x) \in \Omega$ , $X t (x)$ is an element of $T x M$ .

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

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

is nonempty, $X t$ is a vector field in the usual sense defined on the open set $\Omega_t \subset M$ .

[edit] 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.

[edit] Integral curve

An integral curve of the equation above (also called an integral curve of X) is a map

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

such that $\forall t_0 \in I$ , $(t 0,α(t 0))$ is an element of the domain of definition of X and

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

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

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

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

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

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

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

[edit] Flow

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

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

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

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

is the integral curve of X $α$ that verifies $α(t 0) = x$ .

[edit] Properties

We define $F t, s$ as $F t, s (p) = F (t, s, p)$

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

[edit] 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 $η$ 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.

[edit] 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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Category: Differential geometry