Monoparametric Families of Orbits Produced by Planar Potentials
Round 1
Reviewer 1 Report
The manuscript refers to the inverse problem of Newtonian dynamics and offers means to find potentials compatible with a given family of orbits.
For completeness it will be necessary to consider in detail the allowed regions (briefly mentioned in lines 25-26) and to find them in the examples.
As for the integrability of the potentials, it may be exposed in a separate section with a theoretical introduction.
We recommend to display in the figures more orbits in order to emphasize the idea of a family of orbits.
In line 88 it must be 'will not be'.
The quality of the language needs improvements.
There are some spelling mistakes (comemnts - line 64, wriiten - line 178, etc.) and also punctuation ones (comma instead period in (33), (37) etc.).
Author Response
We would like to thank the 1st Reviewer for his fruitful comments.
- The manuscript refers to the inverse problem of Newtonian dynamics and offers means to find potentials compatible with a given family of orbits. For completeness it will be necessary to consider in detail the allowed regions (briefly mentioned in lines 25-26) and to find them in the examples.
Reply: We have done that. We have written a small paragraph at the end of the Introduction and we estimated this region. We indicated the allowed area in each example and those which exist in tables 1,2. Unfortunately, the allowed region is not defined in example 3 (page 6) and we excluded it from our study. The harmonic oscillator allows only for bounded orbits. This topic needs further investigation. The same happened for the second example of integrable potentials (Page 10 of 14), i.e.,
and we removed that. A new example was added in Section 7 (Page 11 of 16).
We thank the Reviewer for this statement.
- As for the integrability of the potentials, it may be exposed in a separate section with a theoretical introduction.
Reply: We have added a new Section, i.e., Section 8 with a small paragraph and we have renamed the Section 8 entitled with “One-dimensional” potentials to Section 9. Furthermore, we replaced the two examples of “one-dimensional” potentials with a new one and we have found one more integrable potential. New references have been added in the bibliography.
- We recommend to display in the figures more orbits in order to emphasize the idea of a familyof orbits.
Reply: We have inserted new figures in the text with families of orbits (Figs. 1, 2, 3. 4).
- In line 88 it must be 'will not be'.
Reply: We have corrected that.
- The quality of the language needs improvements.
There are some spelling mistakes (comemnts - line 64, wriiten - line 178, etc.) and also punctuation ones (comma instead period in (33), (37) etc.).
Reply: We have read the text and we have corrected these grammatical errors.
Other revisions:
Some comments were added at the end of Section 2 after the suggestion of 2nd Reviewer.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
- The slope function gamma=gamma (x, y) is a function of (x, y). this is not used /invoked very clearly in some of the discussions.
-After Eqn. (4) , replace indices-> subindices
-After Eqn. (5): one-> a
-Clarify better what is said in the last paragraph previous to section 3, related to the implications of the constraint capitalGamma=0, based on the necessary related equations in this manuscript.
-CapitalGamma=0 implies that gamma=(gamma/suby)/(gamma/subx).
-How is interpreted the energy of the family of orbits in (7) for the case when capitalGamma=0.
- Line before Proposition 1: “ the next”-> “ the next result”.
- Eqns. 16-18 and joint text: Interpret more clearly by introducing the value of the slope “gamma” into (16)-(17).
- The proof of proposition 1 should be introduces at least as an outline of proof by using/mentioning the previous necessary discussed steps.
- Are both integrals in (18) indefinite since they do not have integration limits?.
- Why P(u) is nonzero for any u ( the argument “u” has different expressions according to the introduction and can be zero if x=0 or if x=y=0 for different definitions of the potential. And also the various constants could be zero as well.
- For instance , in Example 1- Eqn. 22, P(u)=0 if the constants d=sub1=d/sub2=0 .
- P(u)=constant is addressed in the counterexample Example 7. However P(u) being non-constant is not apparently given as a necessary condition in Proposition 1 and Theorem 1. Would it be needed such a necessary condition?.
- The set of references should be extended as well as the related comments within the manuscript.
Author Response
We would like to thank the 2nd Reviewer for his fruitful comments.
- The slope function gamma=gamma (x, y) is a function of (x, y). this is not used /invoked very clearly in some of the discussions.
Reply: We have added some comments in Section 2.
-After Eqn. (4) , replace indices-> subindices
Reply: We have corrected that.
-After Eqn. (5): one-> a
Reply: We have corrected that.
-Clarify better what is said in the last paragraph previous to section 3, related to the implications of the constraint capitalGamma=0, based on the necessary related equations in this manuscript.
Reply: We have added a comment at the end of Section 2.
-CapitalGamma=0 implies that gamma=(gamma/suby)/(gamma/subx)
Reply: We have added a comment at the end of Section 2.
-How is interpreted the energy of the family of orbits in (7) for the case when capitalGamma=0.
Reply: In case Γ=0 , the curves are straight lines and the motion of test particle is not restricted to a specific area. In this case, the energy of family of orbits is not defined as E
- Line before Proposition 1: “ the next”-> “ the next result”.
Reply: We have corrected that.
- Eqns. 16-18 and joint text: Interpret more clearly by introducing the value of the slope “gamma” into (16)-(17).
Reply: We cannot write analytically the expression of the slope function γ in eqs. (16)—(17). It is very complicated.
- The proof of proposition 1 should be introduces at least as an outline of proof by using/mentioning the previous necessary discussed steps.
Reply: We have added a comment at this point.
- Are both integrals in (18) indefinite since they do not have integration limits?.
Reply: Yes, of course. This is done because we integrate twice the relation (18).
- Why P(u) is nonzero for any u ( the argument “u” has different expressions according to the introduction and can be zero if x=0 or if x=y=0 for different definitions of the potential. And also the various constants could be zero as well.
Reply: The function P(u) represents well the potential V=V(x,y). It is not possible to have P(u)=0, because we look for original 2D potentials V=V(x,y) which produce a family of curves as orbits of the test particle. Furthermore, if the function P(u) takes zero values at some points, e.g. at x=0 or at y=0, then we have to clear out this situation from the beginning.
- For instance , in Example 1- Eqn. 22, P(u)=0 if the constants d=sub1=d/sub2=0 .
Reply: We have corrected that. We have written that d_{1}
- P(u)=constant is addressed in the counterexample Example 7. However P(u) being non-constant is not apparently given as a necessary condition in Proposition 1 and Theorem 1. Would it be needed such a necessary condition?.
Reply: We have added a comment after (18).
- The set of references should be extended as well as the related comments within the manuscript.
Reply: We have added new references in the bibliography.
Other revisions:
- We have estimated the allowed region for the motion of test particle in the examples after the suggestion of 1st Reviewer. Unfortunately, the allowed region is not defined in the example 3 (page 6) and we excluded it from our study. The harmonic oscillator allows only for bounded orbits. This topic needs further investigation. The same happened for the second example of integrable potentials (Page 10 of 14), i.e.,
and we removed that. A new example was added in Section 7 (Page 11 of 16).
2) We have added a new Section, i.e., Section 8 with a small paragraph concerning the integrability of planar potentials after the suggestion of 1st Reviewer. Thus, we renamed Section 8 entitled with “One-dimensional” potentials to Section 9. Furthermore, we replaced the two examples of “one-dimensional” potentials with a new one and we have found one more integrable potential.
Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript presents solvable versions of 2-dimensional inverse problem of dynamics. It is an interesting study which focuses on central potentials and extends to homogeneous potentials. The derivations and mathematical correctness are sound. The paper is also written well, I have just one comment/suggestion:
1. I suggest the author consider about presenting some more graphics for other examples solved in the paper for the readers if possible.
Remove words such as 'shall try to' in the manuscript and make it more formal where needed.
Author Response
We would like to thank the 3rd Reviewer for his fruitful comments.
The manuscript presents solvable versions of 2-dimensional inverse problem of dynamics. It is an interesting study which focuses on central potentials and extends to homogeneous potentials. The derivations and mathematical correctness are sound. The paper is also written well, I have just one comment/suggestion:
- I suggest the author consider about presenting some more graphics for other examples solved in the paper for the readers if possible.
Reply: We have added new figures in the text (Figs, 1, 2, 3, 4).
- Remove words such as 'shall try to' in the manuscript and make it more formal where needed.
Reply: We have corrected these grammatical errors.
Other revisions:
- We have estimated the allowed region for the motion of test particle in the examples after the suggestion of 1st Reviewer. Unfortunately, the allowed region is not defined in the example 3 (page 6) and we excluded it from our study. The harmonic oscillator allows only for bounded orbits. This topic needs further investigation. The same happened for the second example of integrable potentials (Page 10 of 14), i.e.,
and we removed that. A new example was added in Section 7 (Page 11 of 16).
2) We have added a new Section, i.e., Section 8 with a small paragraph concerning the integrability of planar potentials after the suggestion of 1st Reviewer. Thus, we renamed Section 8 entitled with “One-dimensional” potentials to Section 9. Furthermore, we replaced the two examples of “one-dimensional” potentials with a new one and we have found one more integrable potential.
3) Some comments were added at the end of Section 2 after the suggestion of 2nd Reviewer.
Author Response File: Author Response.pdf
