Integral curve

In mathematics, an integral curve for a vector field defined on a manifold is a curve in the manifold whose tangent vector (i.e. time derivative) at each point along the curve is the vector field itself at that point. Intuitively, an integral curve traces out the path that an imaginary particle moving in the vector field would follow. Integral curves are closely related to solutions of ordinary differential equations and initial value problems.

Definition

Let "M" be a Banach manifold of class "C"^"r" with "r" ≥ 2. As usual, T"M" denotes the tangent bundle of "M" with its natural projection "π"_"M" : T"M" → "M" given by

:$pi\_\{M\}\; :\; (x,\; v)\; mapsto\; x.$

A vector field on "M" is a cross-section of the tangent bundle T"M", i.e. an assignment to every point of the manifold "M" of a tangent vector to "M" at that point. Let "X" be a vector field on "M" of class "C"^"r"−1 and let "p" ∈ "M". An integral curve for "X" passing through "p" at time "t"₀ is a curve "α" : "J" → "M" of class "C"^"r"−1, defined on an open interval "J" of the real line R containing "t"₀, such that

:$alpha\; (t\_\{0\})\; =\; p;,$:$alpha\text{'}\; (t)\; =\; X\; (alpha\; (t))\; mbox\{\; for\; all\; \}\; t\; in\; J.$

Relationship to ordinary differential equations

The above definition of an integral curve "α" for a vector field "X", passing through "p" at time "t"₀, is the same as saying that "α" is a local solution to the ordinary differential equation/initial value problem

:$alpha\; (t\_\{0\})\; =\; p;,$:$alpha\text{'}\; (t)\; =\; X\; (alpha\; (t)).$

It is local in the sense that it is defined only for times in "J", and not necessarily for all "t" ≥ "t"₀ (let alone "t" ≤ "t"₀). Thus, the problem of proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations/initial value problems and showing that they are unique.

Remarks on the time derivative

In the above, "α"′("t") denotes the derivative of "α" at time "t", the "direction "α" is pointing" at time "t". From a more abstract viewpoint, this is the Fréchet derivative:

:$(mathrm\{d\}\_\{t\}\; f)\; (+1)\; in\; mathrm\{T\}\_\{alpha\; (t)\}\; M.$

In the special case that "M" is some open subset of R^"n", this is the familiar derivative

:$left(\; frac\{mathrm\{d\}\; alpha\_\{1\{mathrm\{d\}\; t\},\; dots,\; frac\{mathrm\{d\}\; alpha\_\{n\{mathrm\{d\}\; t\}\; ight),$

where "α"₁, ..., "α"_"n" are the components of "α" in the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of induced maps. Note that the tangent bundle T"J" of "J" is the trivial bundle "J" × R and there is a canonical cross-section "ι" of this bundle such that "ι"("t") = 1 (or, more precisely, ("t", 1)) for all "t" ∈ "J". The curve "α" induces a bundle map "α"_∗ : T"J" → T"M" so that the following diagram commutes:

:

Then the time derivative "α"′ is the composition "α"′ = "α"_∗ o "ι", and "α"′("t") is its value at some point "t" ∈ "J".

References

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