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 Ω ⊂ R × M {\displaystyle \Omega \subset \mathbb {R} \times M} on T M {\displaystyle TM}
X : Ω ⊂ R × M ⟶ T M ( t , x ) ⟼ X ( t , x ) = X t ( x ) ∈ T x M {\displaystyle {\begin{aligned}X:\Omega \subset \mathbb {R} \times M&\longrightarrow TM\\(t,x)&\longmapsto X(t,x)=X_{t}(x)\in T_{x}M\end{aligned}}}
such that for every ( t , x ) ∈ Ω {\displaystyle (t,x)\in \Omega } , X t ( x ) {\displaystyle X_{t}(x)} is an element of T x M {\displaystyle T_{x}M} . For every t ∈ R {\displaystyle t\in \mathbb {R} } such that the set
Ω t = { x ∈ M ∣ ( t , x ) ∈ Ω } ⊂ M {\displaystyle \Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M}
is nonempty, X t {\displaystyle X_{t}} is a vector field in the usual sense defined on the open set Ω t ⊂ M {\displaystyle \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:
d x d t = X ( t , x ) {\displaystyle {\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
α : I ⊂ R ⟶ M {\displaystyle \alpha :I\subset \mathbb {R} \longrightarrow M}
such that ∀ t 0 ∈ I {\displaystyle \forall t_{0}\in I} , ( t 0 , α ( t 0 ) ) {\displaystyle (t_{0},\alpha (t_{0}))} is an element of the domain of definition of X and
d α d t | t = t 0 = X ( t 0 , α ( t 0 ) ) {\displaystyle {\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 {\displaystyle X} on M {\displaystyle M} can be thought of as a vector field X ~ {\displaystyle {\tilde {X}}} on R × M , {\displaystyle \mathbb {R} \times M,} where X ~ ( t , p ) ∈ T ( t , p ) ( R × M ) {\displaystyle {\tilde {X}}(t,p)\in T_{(t,p)}(\mathbb {R} \times M)} does not depend on t . {\displaystyle t.}
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