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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?

(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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