Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.

Name

Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits.

Definition

Suppose that Template:Math is a static vector field, that is, a vector-valued function with Cartesian coordinates Template:Math, and that Template:Math is a parametric curve with Cartesian coordinates Template:Math. Then Template:Math is an integral curve of Template:Math if it is a solution of the autonomous system of ordinary differential equations,

<math display="block">\begin{align} \frac{dx_1}{dt} &= F_1(x_1,\ldots,x_n) \\ &\;\, \vdots \\ \frac{dx_n}{dt} &= F_n(x_1,\ldots,x_n). \end{align} </math>

Such a system may be written as a single vector equation,

<math display="block">\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).</math>

This equation says that the vector tangent to the curve at any point Template:Math along the curve is precisely the vector Template:Math, and so the curve Template:Math 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.

Examples

File:Slope Field.png

Three integral curves for the slope field corresponding to the differential equation Template:Math.

If the differential equation is represented as a vector field or slope field, then the corresponding integral curves are tangent to the field at each point.

Generalization to differentiable manifolds

Definition

Let Template:Math be a Banach manifold of class Template:Math with Template:Math. As usual, Template:Math denotes the tangent bundle of Template:Math with its natural projection Template:Math given by

<math display="block">\pi_M : (x, v) \mapsto x.</math>

A vector field on Template:Math is a cross-section of the tangent bundle Template:Math, i.e. an assignment to every point of the manifold Template:Math of a tangent vector to Template:Math at that point. Let Template:Math be a vector field on Template:Math of class Template:Math and let Template:Math. An integral curve for Template:Math passing through Template:Math at time Template:Math is a curve Template:Math of class Template:Math, defined on an open interval Template:Math of the real line Template:Math containing Template:Math, such that

<math display="block">\begin{align} \alpha(t_0) &= p; \\ \alpha' (t) &= X (\alpha (t)) \text{ for all } t \in J. \end{align}</math>

Relationship to ordinary differential equations

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

<math display="block">\begin{align} \alpha(t_0) &= p; \\ \alpha' (t) &= X (\alpha (t)). \end{align}</math>

It is local in the sense that it is defined only for times in Template:Math, and not necessarily for all Template:Math (let alone Template:Math). 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, Template:Math denotes the derivative of Template:Math at time Template:Math, the "direction Template:Math is pointing" at time Template:Math. From a more abstract viewpoint, this is the Fréchet derivative:

<math display="block">(\mathrm{d}_t\alpha) (+1) \in \mathrm{T}_{\alpha (t)} M.</math>

In the special case that Template:Math is some open subset of Template:Math, this is the familiar derivative

<math display="block">\left( \frac{\mathrm{d} \alpha_1}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_n}{\mathrm{d} t} \right),</math>

where Template:Math are the coordinates for Template:Math 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 Template:Math of Template:Math is the trivial bundle Template:Math and there is a canonical cross-section Template:Math of this bundle such that Template:Math (or, more precisely, Template:Math) for all Template:Math. The curve Template:Math induces a bundle map Template:Math so that the following diagram commutes:

File:CommDiag TJtoTM.png

Then the time derivative Template:Math is the composition Template:Math is its value at some point Template:Math.

References

• Template:Cite book