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Magnetic Lines of Force

It appears that our computing machines are especially well adapted to the study of properties in the large of the system of lines of magnetic force, due to given currents in space. The renewed interest in the qualitative, ergodic, or even just topological behavior of such families of lines is due to studies in magneto-hydrodynamics, applications in astronomy, questions of origin of the cosmic rays (Fermi), not to mention the importance of such knowledge for applications in the construction of high energy accelerating machines (cyclotrons, synchrotons, etc).

In order to study this subject systematically, it is best to consider first steady currents following through given lines (wires) in space. If there is only one current through a straight line extending infinitely, the system of lines of magnetic force is, of course, very well known. They form circles linking the line; the same is, or course, true for a current in a single closed circle.

However, the topology of the system of lines of force seems to be very complicated and the ergodic behavior of single lines of force unknown for the case where the single closed curve, through which the current flows, forms a knotted loop, say in the simplest case a clover leaf knot. Some single lines of force will probably be dense on two dimensional surfaces, probably some singularities in the field of lines exist, independently of the metric appearance of the knot, but are present necessarily in every topological knot of this sort. There seems to be little hope of obtaining analytically closed expressions describing the system of lines of force.

The situation is complicated in case of two given currents. It is easily seen, in the case given below, that almost all lines of force will be bounded, not closed and each dense on a surface! Let the two currents flow as follows: current 1 on a straight line, say the z axis, current 2 on a circle x² + y² = 1. In general, except at points where the ratio of the two current strengths is rational, a line of force will exhibit an "ergodic" behavior on a surface of a torus.

We propose to investigate, on the computing machines, the properties of lines of force due to two currents-each on a straight line extending to infinity. The two lines are skew. T₁ flows on the y axis, T2 on the line z = d, y = 0.

One is interested, among other things, in the following questions: Do there exist lines which, although not closed, cross a surface of a fixed sphere infinitely many times? Are there lines going arbitrarily far from a fixed point and returning to its neighborhood? Do there exist lines braiding or linking both given wires any number of times?

The computations of such lines of force do not involve much of the "memory" of the machine. The procedure is this: starting at a point (xo, yo, zo) we compute the direction of the magnetic fields, simply adding the two field strengths, given elementarily from each wire; we perform "short" step (Ax, Ay, Az) in the direction of the (constant in time!) force. The computation of this step is done as follows: we calculate a provisional set of increments (Ax)', (Ay)', (Az)' of the variables, in the new position we calculate the new set (Ax)", (Ay)", (Az)"; we then take Ax = ((Ax)' + (Ax)")/2, Ay, = ((Ay)' + (Ay)")/2, Az = ((Az' + (Az)")/2 and proceed anew. In general this way of solving a system of equations dx/X = dy/Y = dz/Z works well. In our case it is seen that each step is computed in the order of 50 milliseconds; to perform, say, 1,000 steps will take of the order of a minute. The idea is now that with the order of 10⁴ steps we shall be able to get

some qualitative information about a single line of force as follows: it seems practical to take each step long enough so that in, say, 50 to 100 steps one complete "loop" can be described around a wire (in positions where it is expected that the lines of force surround the current). One would expect then to obtain a number of "loops" of the order of a few hundred.

The quantities printed as a result of each such calculations could be for instance:

1. The number of "returns" of a line to a given sphere. One simply has to record on the machine the number of times our line crosses the surface of a given sphere.

2. The number of times and the sense in which a line loops the two wires separately and the number of loops surrounding both together. This can be done simply by computing the work done by moving on the line of force, calculating the loops around each wire, as if it alone had current flowing through it. The Gaussian looping coefficients for any two given curves in space can be quickly computed on the machine.

It is convenient to take the length of the "step" along the magnetic field vector to be inversely proportional to the magnetic field strength at that point. In this way the step is proportional to the distance from the wire since the magnetic field is inversely proportional to that distance. The number of steps per looping of the wire is then constant and the step is appropriately shorter at points where the curvature is greater.

It is worthwhile to point out that some integer valued topological invariants may be computed exactly even though we use difference expressions instead of differential ones, and have also round-off errors. This is due to "e-invariance" theorems on simplicial approximations in topology¹ . Also, so to say, the field of "error vectors" is in general "curl-free."