Improved Equations of the Lagrange Top and Examples of Analytical Solutions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript under consideration is a natural extension of the earlier work [1] by the same author, one of the leading world’s experts on classical mechanics. Ref. [1] was devoted to the systematic description of a free rigid body considered as a constrained system, with its dynamics being governed by the variational principle.

The present manuscript extends the analysis of Ref. [1] to the case of a rigid body with one fixed point in a background gravitational field. The developed formalism uncovers completely new features that have not been described in the literature, such as improvements in the equations of motion. The obtained results are very interesting and are expected to be included in new monographs on modern analytic mechanics. I strongly recommend this manuscript for publication in the journal “Particles.”

Comments on the Quality of English Language

The manuscript contains a few grammatical issues.

Author Response

Thank you for the kind report. It is my pleasure that you find the work interesting.

The English was improved. 

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript refers to the motion of a symmetrical top with one point fixed. This is a classical problem in mechanics, already included in a large number of textbooks, analyzed in different ways (using Newtonian or Lagrangian mechanics), and it would have been really surprising if all these where incomplete and need to be corrected as the author claims. 

To study the problem it is convenient to use two coordinate frames. One inertial and another fixed on the rigid body. The coordinates in these frames are connected through the Euler angles and the transformation equation 41 of the manuscript.

Choosing the so-called principal axes as the fixed system on the body, the expressions of angular momentum and kinetic energy are greatly simplified, corresponding to a diagonal inertial tensor. The expression of the kinetic energy corresponds to the two first terms on the right-hand side of equation 42 (proof can be found e.g. in reference 16). Note that this form is independent of the choice of the inertial frame. (Only the values of the Euler angles and their derivatives differ for different choices of inertial frames, not the expression of the kinetic energy.)

Usually the inertial frame is chosen with one of its axis vertical, something that simplifies the gravitational potential energy. The author chooses a different inertial frame, with axes the principal axes of the top at the initial time t=0. This complicates the form of the gravitational potential energy and the whole procedure to find solutions. Unnecessarily in my opinion since there is nothing to be gained in return.

Concluding, I disagree with the author's claim in the comment in page 7 "the attempt to simplify the potential energy will lead, instead of (47),
to a Lagrangian with a complicated expression for the kinetic energy." The kinetic energy has the same simple form if one uses principal axes no matter what is the choice of the inertial frame. And if no flaw in the proof of the related expression (as done e.g. in reference 16) has been found, there is no point to revise this classical textbook problem using a more complicated procedure.

Author Response

Please see attached pdf

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript entitled "Improved equations of the Lagrange top and examples of analytical solutions" by written by author Alexei A. Deriglazov  is a remarkable foray into the research field of the dynamics of a free rigid body, considered as a system with holonomic constraints. The Hamiltonian and Lagrangian formalisms was approached in a familiar way by using the Cartesian coordinates, and also cylindrical and spherical coordinates. The author use this geometric formalism  for the particular case of a Lagrange top. Some improvements of the equations of motion were obtained and several examples of analytical solutions were presented.

The entire manuscript is well written, with a very interesting structure and well adapted to the proposed purposes. All the computations seem to be correct and the obtained results are clearly presented.

I recommend the publication of this paper, but only after the following issues will be addressed and solved:

1. In order to improve the readability of the paper for the beginners, it is indicated to expand the introductory section with one or two sentences that will describe more precisely the previous basic results and the results obtained in this paper.

2. It would be very good to improve the conclusions section with future developments and practical applications of the obtained theoretical results.

3. It is mandatory to put the paper into the MDPI journal format.

Author Response

Thank you for the positive report. Following your suggestions, I slightly extended the Introduction and Conclusion as follows: 

Introduction: 

... of Lagrange top. This is an axially symmetric heavy body with one point of axial axis fixed. To be specific, imagine a conical top whose point of base is fixed in space. The Lagrange top is one of the oldest problems of mathematical physics, which always has been considered as the classical example of the system integrable according to Liouville which, moreover, admits a relatively simple qualitative description on the base of effective potential \cite{Poin,Whit_1917,Mac_1936,Lei_1965,Gol_2000,Grei_2003,Ham_22,Arn_1,Landau_8}.

However, when formulating its variational problem with the diagonal inertia tensor, we observed that potential energy has more complicated form as compared with that assumed in the literature on dynamics of a rigid body.

 

Conclusion: 

This work thereby shows that in the case of a body subject to constraints, their careful analysis is required when deriving the equations of motion. There are other interesting cases that we intend to consider in the future: the equations of a top, wheel and cylinder, with one point restricted to move on a plane without friction.

Reviewer 4 Report

Comments and Suggestions for Authors

Please, find some comments attached below.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

I consider that minor editing of English language is required.

Author Response

Thank you for the positive report. I apologize that you had to correct my English. I corrected all the mistakes according to your list. 

Reviewer 5 Report

Comments and Suggestions for Authors

In this work, the dynamical equations for the Lagrange top is improved - specifically the formulation of the potential energy with a diagonal inertia tensor. The author considered the formulation of the potential energy for the Lagrange top without simplifying the problem.

After careful inspection, I cannot find any mistakes in the analysis by the author. Thus, I am recommending this manuscript for publication.

Comments on the Quality of English Language

A typo needs to be corrected - e.g. 'polinomial' in the last paragraph in Section IV.  Please check and correct similar errors in the manuscript. 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

 

Related to the second point in the reply of the author which apparently is the source of confusion:

In general eigenvalues do not depend on coordinate systems. I_1, I_2 and I_3 are the eigenvalues of a 3x3 second rank tensor. They are numbers that depend only on intrinsic properties of the body, its mass distribution when we talk about the inertial tensor, they do not depend on the coordinate system. Whatever coordinate system one uses as a starting point, when with appropriate rotation move to the principal axes, the transformation of the tensor will result in the same numbers I_1, I_2 and I_3 (although the initial matrix elements I_{ij} and the rotation matrix are different).

For this reason, there is no flaw in the textbooks of classical mechanics (e.g. Ref. 16), and as I wrote in the previous report, the choice of coordinate system by the author only complicates the equations of motion. 

 

Author Response

Please see attached file

Author Response File: Author Response.pdf