Time dependent vector field explained

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 R x M on TM \begin{align} X: \Omega \subset R x 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 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:

(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 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(d \alpha)/(\left.{ dt) }\right|_ =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 R x M, where \tilde{X}(t,p) \in T _{(t,p)} ( R x M) does not depend on t.

Conversely, associated with a time-dependent vector field

X on M is a time-independent one \tilde{X} R x M \ni (t,p) \mapsto \dfrac{\partial}{\partial t} l | _{t} + X(p) \in T _{(t,p)} ( R x M)

on

R x 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 R x \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) \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:(d)/(dt) \left .{ }\right|_ (F^*_ Y_t)_p = \left(F^*_ \left([X_{t_1},Y_{t_1}] + \frac \left .\right|_ 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:(d)/(dt) \left .{ }\right|_ (F^*_ \eta_t)_p = \left(F^*_ \left(\mathcal_\eta_ + \frac \left .\right|_ \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) . Graduate-level textbook on smooth manifolds.