General Three-Body Problem in Conformal-Euclidean Space: New Properties of a Low-Dimensional Dynamical System
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper presents an investigation of the chaotic dynamics of a system made by three bodies interacting among themselves. It is the continuation of a series of works:
1) Gevorkyan, A.S. On reduction of the general three-body Newtonian problem and the curved geometry. J. Phys. Conf. Ser. 2014, 496, 012030.
2) Ayryan, E.A.; Gevorkyan, A.S.; Sevastyanova, L.A. On the Motion of a Three Body System on Hypersurface of Proper Energy. Phys. Part. Nucl. Lett. 2013, 10, 1–8.
3) Gevorkyan, A.S. On the motion of classical three-body system with consideration of quantum fluctuations. Phys. Atomic Nucl. 2017, 80, 358–365.
4) Gevorkyan, A.S. The Three-body Problem in Riemannian Geometry. Hidden Irreversibility of the Classical Dynamical System. Lob. J. Math. 2019, 40, 1058–1068.
5) Gevorkyan, A.S. New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time’s Arrow of Dynamical Systems. Particles 2020, 3, 576-620.
The first part of the paper contains the geometrization of the problem: the dynamical system (obeying to Newton's laws) is converted into a system of free moving objects moving along geodesics in a curved space. This part is not novel, the same material was developed in the papers 1-5 above.
In the second part, the flow of the system in this curved space is investigated with statistical tools in order to disclose the chaotic and irreversible nature of the motion. This part is original.
I sincerely appreciate and admire the effort and the ingenuity of the authors, in this and their previous studies.
Unfortunately, it looks like the authors are unaware of a large body of literature that is critical for their analysis.
The geometrization of the dynamical systems is quite an old topic, it was fostered in the '90 mainly by M. Pettini (Marseille), but several other authors independently worked on that.
Two review works are
[r1] M.Pettini (2007) "Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics" (Springer)
[r2] L. Cassetti, M. Pettini, E.G.D. Cohen (2000) Physics Reports 337, 237
See also
[r3] B. Eckhardt, J.A. Louw and W.-H. Steeb (1986) Austrian Journal of PHysics 39, 331
and
[r4] L. Horwitz et al (2007) Physical Review Letters 98, 234301
Within the geometrical approach, a pivotal role for the understanding of chaotic motion is played by the Jacobi-Levi-Civita equation, which estimates the rate of divergence between nearby geodesics. In a simplified version, but which is exact for two-dimensional systems, this equation writes as an effective oscillator one:
Y''(s) + R/2 Y(s) == 0
where R is the scalar curvature of the curved manifold. A space endowed with negative curvature features the standard hyperbolic dynamics, with exponential separation of trajectories. It was the achievement of Pettini et al, though, to realize that in most systems, including the chaotic ones, R is positive, yet the system may still feature exponential separation of trajectories, provided that R is not constant, through the parametric instability mechanism.
In section 3.2 of the manuscript, the authors euristically replace the chaotic deterministic motion with a phenomenological stochastic term. This allows them to write down an effective Fokker-Planck equation for the evolution of an ensemble of trajectories. The approach followed by the authors is equivalent to the statistical approach carried out by Pettin and coworkers, where the exact but unknown evolution of R(s) is replaced by a statistical approximation, see section 4.1.3 of [r2].
The whole body of research produced by Pettini and collaborators, thus, shows that the claim of the authors is untenable. There is nothing like a "multidimensional, heterogeneous and oriented" time. Actually, the parameter studied by the authors which they arbitrarily call time, has nothing to do with a time.
The equations of motion do not depend upon the geometry employed, and, both in the Hamiltonian and Riemaniann framework, are formally reversible since symmetrical with respect to both the physical and the proper time. Irreversibility is introduced in the system in the usual way, i.e., through exponential sensitivity to initial conditions in the Jacobi-Levi-Civita equation.
In summary, I think that the paper is still worth publication but it has to be seriously reconsidered, taking into account the huge body of results so far obtained in the geometrization of dynamicsl systems.
I have also some lesser critical, but still important comments.
1) In this and their previous works, the authors persist in a habit that I find pointless. Apart for the two futher symmetries discovered by the authors themselves, the system studied by the authors has 11 constants of motion, not 10 as the authors persist in claiming: the energy is a constant of the motion since no external forces are involved, and the three bodies interact through conservative forces. Furthermore, when the authors geometrize the dynamics, they employ the Jacobi metric. The metric element g_ij depends from the energy of the system, which therefore MUST be a constant. Hence, not only the authors' habit is misleading, it is also plainly wrong in this case.
2) The term "three body problem" is a generic label, insofar one does not explicit the forces interacting between the bodies. Not all three-body systems need being chaotic. In the paper, the inter-body forces employed are introduced at section 5, and are a kind of short-ranged intermolecular forces. I think that it would beneficial to the reader move the description of the forces employed somewhere in the first part of the paper.
Author Response
Please fined the attached file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for Authorssee comments
Comments for author File: Comments.pdf
Author Response
Please find the attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsSee my attached report
Comments for author File: Comments.pdf
Author Response
I am sending pdf file with the answers written to reviewer 1.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for Authorssee comments
Comments for author File: Comments.pdf
Author Response
I am sending a pdf file with the answers written to reviewer 2.
Author Response File: Author Response.pdf
Round 3
Reviewer 2 Report
Comments and Suggestions for Authorssee comments
Comments for author File: Comments.pdf
Author Response
Answers can be found in the attachment.
Author Response File: Author Response.pdf
