Evolving vector field of velocities on a circle

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  • #1
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Summary:

Probably, to specify the evolving vector field of velocities on a circle, one could consider a vector field on a cylinder, where the linear coordinate of the cylinder serves as an evolutionary parameter (logarithm of time), but I would like to set it on a plane so that time is equal to the radius of the circle. How should I do it?
The simplest example of such a construction is a linear vector field
(y+x)\partial_{x} + (y-x)\partial_{y}
in which the tangent to the circle (static) component
y\partial_{ x} - x\partial_{ y}
and the radial to the circle (evolutionary) component
x\partial_{ x} + y\partial_{ y}
are easily separated.
And how to define the evolving vector field of velocities (and accelerations) in the general case?

However, it seems that in the case of a plane the solution is obvious - it is enough to take the tangent component of a linear vector field with a coefficient in the form of a smooth function of the coordinates of a point on the plane, and the radial component with a coefficient as a function of the length of the radius vector of a point on the plane. With evolving vector fields on odd-dimensional spheres, it will probably be possible to do the same. But how to describe the evolving vector field of velocities (with singularities) on the classical 2-dimensional sphere?
 
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Answers and Replies

  • #2
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Unfortunately, I can no longer edit the main text.

The simplest example of such a construction is a linear vector field
$$(y+x)\partial_{x} + (y-x)\partial_{y}$$
in which the tangent to the circle (static) component
$$y\partial_{ x} - x\partial_{ y}$$
and the radial to the circle (evolutionary) component
$$x\partial_{ x} + y\partial_{ y}$$
are easily separated.
 
  • #3
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1
Let's try to construct in 4-dimensional Euclidean space an arbitrary vector field tangent to the hyperspheres of this space. First of all, using Pauli matrices, we construct a linear vector field
$$
\begin{equation}
\begin{split}
\mathrm{i}\sigma_1: \qquad x_4\partial x_1 - x_3\partial x_2 + x_2\partial x_3 - x_1\partial x_4,\\
\mathrm{i}\sigma_2: \qquad x_3\partial x_1 + x_4\partial x_2 - x_1\partial x_3 - x_2\partial x_4,\\
\mathrm{i}\sigma_3: \qquad x_2\partial x_1 - x_1\partial x_2 - x_4\partial x_3 + x_3\partial x_4
\end{split}
\end{equation}
$$
Then we define three components of an arbitrary tangent vector ##v##
$$
\begin{equation}
\begin{split}
v_1=f_1(x)\mathrm{i}\sigma_1,\\
v_2=f_2(x)\mathrm{i}\sigma_2,\\
v_3=f_3(x)\mathrm{i}\sigma_3
\end{split}
\end{equation}
$$
Now one could demand that this vector field obey the principle of minimality, that is, that it be potential, and its level surfaces are minimal surfaces in ##\mathbb{R}^4##. Then the evolution of this vector field could be tracked by changing the radius of the hypersphere.
 
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