Utente:Grasso Luigi/sandbox4/Curva integrale

Tre curve integrali per un campo direzionale dell'equazione differenziale dy / dx = x² − x − 2.

In matematica, una curva integrale è una curva parametrica che rappresenta una soluzione particolare di un'equazione differenziale ordinaria o sistemi di equazioni. Se l'equazione differenziale viene rappresentata come un campo vettoriale o campo direzionale, allora le curve integrali corrispondenti sono tangenti al campo in ogni punto.

Le curve integrali sono conosciute con vari altri nomi, a seconda della natura e dell'interpretazione dell'equazione differenziale o del campo vettoriale. In fisica, le curve integrali per un campo elettrico o un campo magnetico sono conosciute come linee di campo, e quelle di un campo velocità di un fluido sono dette linee di flusso. Nei sistemi dinamici, le curve integrali per l'equazione differenziale che descrive un sistema sono dette traiettorie o orbite.

Definizione

Suppose that F is a vector field: that is, a vector-valued function with Cartesian coordinates (F₁,F₂,...,F_n); and x(t) a parametric curve with Cartesian coordinates (x₁(t),x₂(t),...,x_n(t)). Then x(t) is an integral curve of F if it is a solution of the following autonomous system of ordinary differential equations:

${\begin{aligned}{\frac {dx_{1}}{dt}}&=F_{1}(x_{1},\ldots ,x_{n})\\&\vdots \\{\frac {dx_{n}}{dt}}&=F_{n}(x_{1},\ldots ,x_{n}).\end{aligned}}$

Such a system may be written as a single vector equation

$\mathbf {x} '(t)=\mathbf {F} (\mathbf {x} (t)).\!\,$

This equation says that the vector tangent to the curve at any point x(t) along the curve is precisely the vector F(x(t)), and so the curve x(t) is tangent at each point to the vector field F.

If a given vector field is Lipschitz continuous, then the Picard–Lindelöf theorem implies that there exists a unique flow for small time.

Generalizzazione a varietà differenziabili

Definizione

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

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

A vector field on M is a cross-section of the tangent bundle TM, 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 t₀ is a curve α : J → M of class C^r−1, defined on an open interval J of the real line R containing t₀, such that

$\alpha (t_{0})=p;\,$

$\alpha '(t)=X(\alpha (t)){\mbox{ for all }}t\in J.$

Relazione con equazioni differenziali ordinarie

The above definition of an integral curve α for a vector field X, passing through p at time t₀, is the same as saying that α is a local solution to the ordinary differential equation/initial value problem

$\alpha (t_{0})=p;\,$

$\alpha '(t)=X(\alpha (t)).\,$

It is local in the sense that it is defined only for times in J, and not necessarily for all t ≥ t₀ (let alone t ≤ t₀). 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.

Osservazioni sulla derivata temporale

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 Rⁿ, this is the familiar derivative

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

where α₁, ..., α_n are the coordinates for α with respect to the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of induced maps. Note that the tangent bundle TJ 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_∗ : TJ → TM 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.

Note

• Serge Lang, Differential manifolds, Reading, Mass.–London–Don Mills, Ont., Addison-Wesley Publishing Co., Inc., 1972.

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