Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems

Round 1

Reviewer 1 Report

This work under review, the author study foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. A large number of results (with applications and examples) are presented regarding this field. 

Comments: (i) Labeled the system given by equation (5) and (6) in a single equation.

(ii) The first section become too large, so if possible please reduce the volume of Section-1 and 2 by truncating the definition of obvious terms.

(iii) write the equation (80) properly.

(iv) Sometimethe symbols are not properly expressed.

Author Response

Concerning: "(i) Labeled the system given by equation (5) and (6) in a single equation." - I improved this.

Concerning: "(ii) The first section become too large, so if possible please reduce the volume of Section-1 and 2 by truncating the definition of obvious terms." - I reduced the Sect. 1 by transferring two subsections into Sect.2. Since the work is a short survey for theoretical physics community, almost all content can be considered as a trivial by an expert. But I believe that the introductory sections are important to make the survey a self contained and comfortable for the beginners, and I suggest to keep them in the present form. 

Concerning: "(iii) write the equation (80) properly." - I improved this (and other) equations, all now fit in the lines.

 

 

 

 

Reviewer 2 Report

I’m sorry but I’ll not be able to complete the review of the manuscript universe-1915547.

I agreed to review an article, but this manuscript is not formatted as a research paper; it is more similar to a chapter of a book. It contains a general exposition of Poisson and symplectic geometry, but there are not (as far as I can see)– original results.

-In Section 1 a clear structure of the paper is missing.

-Several ‘exercises’ are proposed in the manuscript.

-Conclusions are missing.

-The more recent reference, except one paper of the author, is thirty years old.

Author Response

Concernig: "I agreed to review an article, but this manuscript is not formatted as a research paper; it is more similar to a chapter of a book. It contains a general exposition of Poisson and symplectic geometry"; "Several ‘exercises’ are proposed in the manuscript."   - 

According to “Aims/Scope” of the Universe: "It publishes reviews, research papers and communications." When I was invited to submit the work to the Universe by the editorial board, during the preliminary correspondence I clearly explained that the work is a short review, that at the same time could be useful for the beginners in the field of theoretical and mathematical physics. 

Concerning: "In Section 1 a clear structure of the paper is missing." - The list of "contents" was excluded during the formation process by editorial board, now it is restored. 

Concerning:  "Conclusions are missing." - I added the section "Conclusions" . 

Concerning: "The more recent reference, except one paper of the author, is thirty years old"- the list of references was essentially extended. 

Concerning: " there are not (as far as I can see)– original results." - while the work is a survey, I discussed geometrical meaning and construction of the Dirac bracket, and some  applications. These points (as far as I know) are not presented in the theoretical-physics literature. The results are discussed in the "Conclusion" of revised version. 

 

 

Round 2

Reviewer 2 Report

I think the manuscript has been now sufficiently improved to be published in Universe.

I only recommend translating into English the few Portuguese words still present in the manuscript:

-<<sumário>> (p. 1, line 9);

-<<Version 10 de outubro de 2022>> (at the beginning of every page).