Let

S=φ=∫ω(t)dt

where

ω=dφ/dt

Then a variation of the action

δ∫ω(t)dt=0

leads to the differential equation

d2φ/dt2=0

which entails

ω=dφ/dt=const

Next, let the motion of the circle be set in evolution so that the radius of the circle is related to the evolutionary parameter of time t by dependence

at=log(ρ)

Vr=dρ/dt=aρ

Vr=a(x∂x + y∂y)

In turn,

ω=dφ/dt=Vτ/ρ=b

Vτ=b(y∂x - x∂y)

Thus, due to the correspondence of the algebra of complex numbers and the resulting algebra of linear vector fields, we have

1= x∂x + y∂y

i = y∂x - x∂y

v = a + bi = c

Follow the link below

Article Applications of the local algebras of vector fields to the m...

If each point of the plane serves as the center of a rotating circle, then each point corresponds to a local algebra of vector fields isomorphic to the field of complex numbers. In this case, only conformal transformations of the plane preserve local algebras of vector fields, which in turn induce analytic transformations of the complex plane.

For those who are waiting for the continuation, let me offer a fragment of the book at the link

https://www.researchgate.net/publication/377842836_Mathematical_Notes_on_the_Nature_of_Things_fragment

We offer you a collection of mathematical notes by the author, which are designed to serve as a tool for describing the nature of things. Thus, the subject of this book is the mathematical formalism of the theory of everything, and as the fundamental idea of the book, the principle of least action is used, raised to a geometric-dynamic level. The book is intended primarily for those readers who are interested in the metaphysical foundation of physics, i.e., questions about the origin of the laws of physics, physical fields and particles, and finally, about the origin of physical space and the dependence of the mathematical description of physical space on the localization of the observer. The content of the main part of the book consists of an introduction and five notes revealing the philosophical intent of the author, and in the appendix there are two more notes of an applied nature. It should also be noted that this small brochure is addressed not only to readers who are able to get into the philosophical and mathematical idea of the author, but also to skeptical readers who may be interested only in the mathematical aspects of this idea.

Dear Igor Bayak ,

is there any similarity of your proposed topology, with the line element field described by Nash: Gary Nash

Preprint Modified General Relativity and dark energy