Integral Curve

• An integral curve is a line that, at each point, is tangent to a vector field

• The problem of finding an integral curve can be viewed as a problem of integrating a given continuous function

  • Intuition: The integral curve of a vector field in is a graph in . Each point in has an associated slope, i.e., an associated derivative. The fundamental theorem of calculus then suggests we can find the integral curve via integration

• A necessary and sufficient condition for the graph of function to be an integral curve is that the following relation hold for interval

  The integral line lies in the direct product of the time axis and phase space (called the extended phase space)

  • Integral Curves are solutions to an ODE.
  • In fact Every first order ODE determines a direction field on the plane. At , the slope is .
  • Singular points also called Equilibrium points correspond to points where vanishes. That is, when
    At such points, we have constant functions of the form as solutions.

• (Arnold 2.1.1) Consider a differential equation of the form

  Where is a smooth function defined on . That is is continuously differentiable.

  A solution satisfying the above satisfying the initial condition can be found as follows

  Note that this also gives an integral curve for . The first equation is called Barrow’s Formula

  Such a solution exists for all and . Such a solution is also unique in the sense that any two solutions with the same condition coincide in some neighborhood to point .

  • It is the smoothness of which guarantees the uniqueness of the solution

  • Intuition: It is easy to show that Barrow’s Formula holds. What is tricky is showing the uniqueness in the case of being an equilibrium point.

    Let be an equilibrium position. By definition we have that .

    Consider an instant infinitesimally close to such that . We now consider the motion from to . Observe two things:

    First, that is smooth and therefore Lipschitz continuous so it is bounded. Second that the motion is locally linear since we assume to be infinitesimally close to .

    Now use Barrow’s formula to find the time it will take to reach from a point . This time, as it turns out, is infinite!

    In other words, it takes an infinitely long time to reach the equilibrium position unless we start from the equilibrium position itself.

• (Arnold 2.4.1) Let

  Be a system of ODEs.

  The solution of the ODE is a mapping of the form where and are solutions of each individual ODE.

  That is, if we have a system of ODEs corresponding to that of integral curves, then we can find the solution by solving each independently and concatenating the results.

  Note that this holds, provided each of the ’s are independent of each other. That is, is only dependent on .

• An equation with separable variables is an equation of the form

  Where and are smooth and do not vanish under the region of interest.

  • (Arnold 2.6.1) Consider the system
    A curve is a phase curve of the above if and only if it is also the integral curves of the equation on separable variables
  • (Arnold 2.6.2) A solution to the above with initial conditions exists, is unique (i.e., two solutions coincide when both are defined), and is given by