Round 1

Reviewer 1 Report

p. 2, l. 44: the calculations may generate new results, however it's not clear that the methodology to produce these results is in the mathematical context new.

Author Response

We would like to thank the 1^st Reviewer for his fruitful comments.

1. 2, l. 44: the calculations may generate new results, however it's not clear that the methodology to produce these results is in the mathematical context new.

Reply: The methodology is not completely new because a similar one was used by the same author in a previous paper (Kotoulas, 2022b) to find two—parametric families of orbits produced by 3D galactic—type potentials. In this paper we considered potentials of the form V=A(x^2+p*y^2+q*z^2) and we restricted ourselves to the case n=2.

The present study is more general. We consider potentials of the form V=U(x^p+y^p+z^p), where p=integer, because we would like to study potentials for p>=2. An application of our method was done for the 3D harmonic oscillator and new results were found.

We have corrected this point as follows: “our methodology…”

Major revisions in the text:

• We added three new paragraphs concerning central and polynomial potentials in the “Introduction” and a small paragraph concerning the new results of this paper in the Section “Conclusions” after the suggestion of 3^rd New references were added in the text and in the bibliography.
• We added a small paragraph for the allowed region of motion of test particle at the end of Section 2 (The basic equations) and we wrote a comment in each example.
• Section 4 was divided into 4 (four) parts: 4.1) Central potentials, 4.2) Polynomial potentials, 4.3) Potentials depending on distance $r$ and 4.4) Polynomial potentials as solutions to the Laplace’s equation.
• In Section 6: We changed the Plan A_1 and the corresponding examples. Instead of two examples, we wrote one new and Fig. 3a was replaced by a new figure in which we show closed orbits produced by the 3D harmonic oscillator.
• In Section 6: We deleted the Plan A_3 and the corresponding example and, finally, we summarized our results.
• In Section 6: We removed Example 6 after the suggestion of the 2^nd Reviewer. The family of orbits f(x,y,z)= is unbounded as x. But the harmonic oscillator ) allows for bounded orbits only. Indeed, we estimated the allowed area for the motion of test particle (Anisiu 2005, pp. 548--550) and we ascertained that this area is not defined in this case. This topic needs further investigation. So, we removed this result from our study and we checked again the other results concerning the harmonic oscillator. We removed only them in which the family of orbits corresponds to open curves. Furthermore, in the Example 7 we gave more explanations about the values of $m$ (m=-2, 0, 2). 
• In Section 7: We removed Plan B_2 and the corresponding example. Finally, the Plan B_3 was renamed to Plan B_2. 

Author Response File: Author Response.docx

Reviewer 2 Report

See pdf file attached

Comments for author File: Comments.pdf

Author Response

We thank the 2^nd Reviewer for his useful comments.

There are serious problems with Example 6 and 7.

• In Example 6, the orbit, defined through f(x,y,z)= is described by the vector

{x,c₁ x,c₂x}, x > 0, which is unbounded for x → ∞. However, if the energy is finite, then the harmonic potential V = (1/2)(4x² + y² + 4z²) allows for bounded orbits only. Since no calculation error is found, the problem appears to be a deeper one and may lie in Proposition 1, which is neither proved in the manuscript nor provided with a reference. Also, the equivalence of the conditions (11) and (17) appear to be not evident.

Reply: The referee is right. The family of orbits f(x,y,z)= is unbounded as x. But the harmonic oscillator ) allows for bounded orbits only. There is no problem with the Proposition 1. The reason is the following:  we estimated the allowed area for the motion of test particle (Anisiu 2005, 548—550) and we ascertained that this area is not defined in this case. The result in Example 6 is doubtful and needs further investigation. So, we removed this result from our study and we checked again the other results concerning the harmonic oscillator. We removed only them in which the family of orbits corresponds to open curves. (The allowed area for the motion of the test particle is given by the type:  . We wrote a comment in Section 2 after the basic equations of the Inverse problem of Dynamics).

• In Example 7, the evaluation of Eq.(11) by Mathematica gives the two cases m = 0 and m = −2, rather than m = 2 and m = −2. More problematic is the claim that R = 0, since both r₁ and r₂ are identically zero for all exponents m with the consequence that R = −r₁/r₂ is not defined.

Reply: We checked again our results and we have the evaluation of eq. (11) by Mathematica provides us with three cases: m=-2, 0 and 2. The case m=2 leads to the well-known harmonic oscillator  and it is not taken into account at this time because we would like to find an expression for the perturbed harmonic oscillator. The case m=0 leads to the case r_1=r_2=0 and the function R=- in eq. (22)  is not defined.  Thus, according to the Case 2 (page 4), we can consider that any function  , A: arbitrary function of C^2-class is solution to our problem. We excluded this value from our study because we do not have perturbation in the harmonic oscillator in this case. So, we left with the value m=-2. We added a comment in the corresponding paragraph.

Publication of the manuscript is not recommended before the two problems are addressed.

Major revisions in the text:

• We added three new paragraphs concerning central and polynomial potentials in the “Introduction” and a small paragraph concerning the new results of this paper in the Section “Conclusions” after the suggestion of 3^rd New references were added in the text and in the bibliography.
• We added a small paragraph for the allowed region of motion of test particle at the end of Section 2 (The basic equations) and wrote a comment in each example.
• Section 4 was divided into 4 (four) parts: 4.1) Central potentials, 4.2) Polynomial potentials, 4.3) Potentials depending on distance $r$ and 4.4) Polynomial potentials as solutions to the Laplace’s equation.
• In Section 6: We changed the Plan A_1 and the corresponding examples. Instead of two examples, we wrote one new and Fig. 3a was replaced by a new figure in which we show closed orbits produced by the 3D harmonic oscillator.
• In Section 6: We deleted the Plan A_3 and the corresponding example and finally, we summarized our results.
• In Section 6: We removed Example 6. As we explained above, the family of orbits f(x,y,z)= is unbounded as x. But the harmonic oscillator ) allows for bounded orbits only. Indeed, we estimated the allowed area for the motion of test particle (Anisiu 2005, pp. 548--550) and we ascertained that this area is not defined in this case. This topic needs further investigation. So, we removed this result from our study and we checked again the other results concerning the harmonic oscillator. We removed only them in which the family of orbits corresponds to open curves. Furthermore, in the Example 7 we gave more explanations about the values of $m$ (m=-2, 0, 2). 
• In Section 7: We removed Plan B_2 and the corresponding example. Finally, the Plan B_3 was renamed to Plan B_2. 

Author Response File: Author Response.docx

Reviewer 3 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

We would like to thank the 3^rd  Reviewer for his fruitful comments.

Major revisions in the text:

• We added three new paragraphs concerning central and polynomial potentials in the “Introduction” and a small paragraph concerning the new results of this paper in the Section “Conclusions” after the suggestion of 3^rd New references were added in the text and in the bibliography.
• We added a small paragraph for the allowed region of motion of test particle at the end of Section 2 (The basic equations) and we wrote a comment in each example.
• Section 4 was divided into 4 (four) parts: 4.1) Central potentials, 4.2) Polynomial potentials, 4.3) Potentials depending on distance $r$ and 4.4) Polynomial potentials as solutions to the Laplace’s equation.
• In Section 6: We changed the Plan A_1 and the corresponding examples. Instead of two examples, we wrote one new and Fig. 3a was replaced by a new figure in which we show closed orbits produced by the 3D harmonic oscillator.
• In Section 6: We deleted the Plan A_3 and the corresponding example and, finally, we summarized our results.
• In Section 6: We removed Example 6 after the suggestion of the 2^nd The family of orbits f(x,y,z)= is unbounded as x. But the harmonic oscillator ) allows for bounded orbits only. Indeed, we estimated the allowed area for the motion of test particle (Anisiu 2005, pp. 548) and we ascertained that this area is not defined in this case. This topic needs further investigation. So, we removed this result from our study and we checked again the other results concerning the harmonic oscillator. We removed only them in which the family of orbits corresponds to open curves. Furthermore, in the Example 7 we gave more explanations about the values of $m$ (m=-2, 0, 2). 
• In Section 7: We removed Plan B_2 and the corresponding example. Finally, the Plan B_3 was renamed to Plan B_2. 

Author Response File: Author Response.docx

Reviewer 4 Report

Comments for author File: Comments.pdf

Author Response

Referee report for Manuscript ID:

Axioms-2307550

April 2, 2023

We thank the 4^th  Reviewer for his useful comments.

Dear Editor,                                                                                                                                   

I have read with interest the manuscript Axioms-2307550. The author studies three-dimensional potentials which produce a preassigned twoparametric family of spatial regular orbits given in the solved form f(x, y, z) = c₁, g(x, y, z) = c₂ (c₁, c₂ = const). While these potentials have to satisfy two linear PDEs which are the basic equations of the 3D Inverse Problem of Newtonian Dynamics. The author offers pertinent examples of potentials that are mainly used in physical problems. The author also founds new families of orbits produced by the 3D harmonic oscillator along with pertinent examples are given and cover all the cases. The author performs a great deal of work in these kinds of three-dimensional central and polynomial potentials. However, I have some major concerns with this paper that the author should consider before any consideration.

1./ The references and discussion in the introduction are all rather general and do not explain well why these three-dimensional central and polynomial potentials are important, to which experiments it relates to closely, or what other studies of these three-dimensional central and polynomial potentials have done. Or to what degree it has been solved and reported on previously.

Reply: We have added a small paragraph in the Introduction for central and polynomial potentials and added new references in the bibliography.

2./ The author should state in the introduction the main purpose of the work.

Reply: We have added a small paragraph in the Introduction

1

3./ Unfortunately, the authors are not aware of any such works. There are no tangible factual comparisons to other works, so it is difficult to say how relevant or novel this work is.

Reply: We have added a small paragraph in the Introduction.

4./ The summary should be written more strongly than that, in which the important results and the most important parameters affecting the results should be presented.

Reply: We have added a small paragraph in the Section “Conclusions”.

After these major improvements, the paper can be reconsidered.

Major revisions in the text:

• We added three new paragraphs concerning central and polynomial potentials in the “Introduction” and a small paragraph concerning the new results of this paper in the Section “Conclusions” after the suggestion of 3^rd New references were added in the text and in the bibliography.
• We added a small paragraph for the allowed region of motion of test particle at the end of Section 2 (The basic equations) and wrote a comment in each example.
• Section 4 was divided into 4 (four) parts: 4.1) Central potentials, 4.2) Polynomial potentials, 4.3) Potentials depending on distance $r$ and 4.4) Polynomial potentials as solutions to the Laplace’s equation.
• In Section 6: We changed the Plan A_1 and the corresponding examples. Instead of two examples, we wrote one new and Fig. 3a was replaced by a new figure in which we show closed orbits produced by the 3D harmonic oscillator.
• In Section 6: We deleted the Plan A_3 and the corresponding example and, finally, we summarized our results.
• In Section 6: We removed Example 6 after the suggestion of the 2^nd The family of orbits f(x,y,z)= is unbounded as x. But the harmonic oscillator ) allows for bounded orbits only. Indeed, we estimated the allowed area for the motion of test particle (Anisiu 2005, pp. 548—550) and we ascertained that this area is not defined in this case. This topic needs further investigation. So, we removed this result from our study and we checked again the other results concerning the harmonic oscillator. We removed only them in which the family of orbits corresponds to open curves. Furthermore, in the Example 7 we gave more explanations about the values of $m$ (m=-2, 0, 2). 
• In Section 7: We removed Plan B_2 and the corresponding example. Finally, the Plan B_3 was renamed to Plan B_2. 

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The revised manuscript is recommended for publication

Reviewer 4 Report

The author has addressed all the relevant issues raised. Now the paper deserves to be accepted for publication.