Vector field on-the circle

These general results are fundamentally topological in origin. They reflect the fact that X = f(x corresponds to flow on a line. If you flow monotonically on a line, you ll never come back to your starting place—that s why periodic solutions are impossible. (Of course, if we were dealing with a circle rather than a line, we could eventually return to our starting place. Thus vector fields on the circle can exhibit periodic solutions, as we discuss in Chapter 4.)... 

Let s begin with some examples, and then give a more careful definition of vector fields on the circle. 

Example 4.1.2 suggests how to define vector fields on the circle. Here s a geometric definition A vector field on the circle is a rule that assigns a unique velocity vector to each point on the circle. 

In practice, such vector fields arise when we have a first-order system 0 = /(0), where /(0) is a real-valued, 27t-periodic function. That is, f d + 2it) = /(0) for all real 0. Moreover, we assume (as usual) that f(Q) is smooth enough to guarantee existence and uniqueness of solutions. Although this system could be regarded as a special case of a vector field on the line, it is usually clearer to think of it as a vector field on the circle (as in Example 4,1,1). This means that we don t distin-... 

The same information can be shown by plotting the vector fields on the circle (Figure 4.3.3). 

For which real values of a does the equation 0 = sin(zz6>) give a well-defined vector field on the circle ... 

In Exercises 2.6.2 and 2.7.7, you were asked to give two analytical proofs that periodic solutions are impossible for vector fields on the line. Review these arguments and explain why they don t carry over to vector fields on the circle. Specifically which parts of the argument fail ... 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

If we try to avoid this non-uniqueness by restricting 0 to the range -zr < 0 < zr, then the velocity vector jumps discontinuously at the point corresponding to 0 = zr. Try as we might, there s no way to consider 0 = 0 as a smooth vector field on the entire circle. 

For each of the following vector fields, find and classify all the fixed points, and sketch the phase portrait on the circle. 

Since the system is 27r-periodic in 0, it may be considered as a vector field on a cylinder. (See Section 6,1 for another vector field on a cylinder.) The x-axis runs along the cylinder, and the 0-axis wraps around it. Note that the cylindrical phase space is finite, with edges given by the circles x = 0 and x = 1. 

A further important property of synchrotron radiation concerns its polarization characteristics. The radiation is completely polarized, and the kind of polarization depends on the direction of the circulating electron beam as well as on the direction of photon emission. In order to understand these polarization properties, it is useful to recall the result for the emission of electromagnetic radiation from an electron moving with non-relativistic velocity in a circle the electric field vector follows the same shape and orientation as the projection of the electron s path onto a plane perpendicular to the observation direction. 

The vast majority of crystals are anisotropic, that is, their properties are not the same in all directions within the crystal. Light is said to be plane polarized (also called linearly polarized) when the electric field oscillates in a straight line. When, on the other hand, the electric field vector travels aronnd a circle, then the light is said to have circnlar polarization, and when the ends of the vector travel in an ellipse, it is said to be elliptically polarized. 

Inasmuch as the vector representing the magnetic field lies in a longitudinal plane and as a consequence of the axial symmetry, a vortex electric field, arising as a result of the change of this magnetic field with time, has but one component E. The vector lines of the field are therefore circles centered on the z-axis. 

In conclusion, it should be emphasized that the vector lines of the electric field are circles lying in horizontal planes with centers located on the z-axis. It is an easy matter to show that the electric field given in eq. 1.205 is practically identical to that of a magnetic dipole (eq. 1.197) when the distance of the observation point to the source is significantly larger than the radius of the loop. 

If the field excitation is realized by vertical magnetic dipole sources of the secondary field are induced currents vector lines of which are located in horizontal planes and they present themselves as circles with centers on the borehole axis. 

FIG. 11.4. (a) Angular momentum caused by an orbiting particle and permitted value projected on the external field axis. The vector p, processes around the field axis as indicated by the dashed circle (ell se in the drawing), (b) The l-s coqtling of orbital angular momentum and spin leads to a resultant angular momentum pj. 

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