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

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A time dependent vector field on a manifold M is a map from an open subset   on  

 

such that for every  ,   is an element of  .

For every   such that the set

 

is nonempty,   is a vector field in the usual sense defined on the open set  .

Associated differential equation

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Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:

 

which is called nonautonomous by definition.

Integral curve

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

 

such that  ,   is an element of the domain of definition of X and

 .

Equivalence with time-independent vector fields

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A time dependent vector field   on   can be thought of as a vector field   on   where   does not depend on  

Conversely, associated with a time-dependent vector field   on   is a time-independent one  

 

on   In coordinates,

 

The system of autonomous differential equations for   is equivalent to that of non-autonomous ones for   and   is a bijection between the sets of integral curves of   and   respectively.

Flow

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

 

such that for every  ,

 

is the integral curve   of X that satisfies  .

Properties

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We define   as  

  1. If   and   then  
  2.  ,   is a diffeomorphism with inverse  .

Applications

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Let X and Y be smooth time dependent vector fields and   the flow of X. The following identity can be proved:

 

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

 

This last identity is useful to prove the Darboux theorem.

References

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  • Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.