Talk:Stability of Hamiltonian equilibria
I have just a few minor comments which hopefully can improve the readability of the article.
1) In Figure 2 mark the positions of z_1, z_2;
2) equation (6): c_j (the index is j) must be defined ("for some real constants...");
3) "Moreover, ±σ,±σ^∗ all have the same multiplicity" tell what the asterisk means (complex conjugate);
4) in the paragraph "Sturm's theorem" the quantities Q must be defined; same for G_k in equation (18)and I would avoid to use t as a subindex;
5) formula (20): tr --> Tr (as for the trace in the rest of the paper);
6) the sentence "A very important example of a non-natural flow is the restricted three-body problem" needs more details;
7) 3 lines after the previous sentence: tr --> Tr .
Responses to Reviewer 1: James Meiss
1) Fixed 2) Fixed 3) Fixed 4) Fixed--re-ordered things to make the sequence more clear, and changed t to K. 5) Fixed 6) Fixed--added a remark about Coriolis forces 7) Fixed.
It seams to me that the article does not really presents a modern vewpoint to the problem of Stability of Hamiltonian Flows because the author writes only from the viewpoint of his personal domain of interest. Here I mean the following. In the article only linear aspect of the problem is essentially discussed. However even here there is at least one gap: the effect of gyroscopic stabilization is not mentioned properly. Nonlinear aspect is touched just superficially: I would expect in the article a discussion of the role of KAM-theory in stability results, a discussion of the Arnold diffusion, and a more detailed discussion of application of Lyapunov functions as a tool for nonlinear case. The title of the article apparently assumes discussion of stability for periodic solutions. If the author does not plan to present all these discussions, he should change title of the article.
I was also surprised by a strange style of citation. References to many classical standard (and in many cases trivial) facts are given not to textbooks but to some journal papers. Here are some details.
"It is not difficult to show that the eigenvalues of a Hamiltonian matrix come in pairs ..." This is a standard fact. The reference to a paper by Mackay looks strange: any textbook, (for example, Arnold) is more relevant.
"The precise relation is that an equilibrium is linearly stable if and only if ..." This statement certainly should be mentioned. But it is trivial and well known for more than 100 years. So reference to Mackay 1986 looks strange.
Nonlinear stability DOES NOT IMPLY linear stability, because linear instability, generated by a non-trivial Jordan block, can be stabilized by nonlinear terms.
"Moreover, $\pm\sigma,\pm\sigma^*$ all have the same ..." As far as I know, this is a corollary from the Williamson theorem (the very beginning of XX century). Why Mackay 1986? Exposition of the Williamson theorem is presented in the Arnold's textbook.
"... can lose spectral ..." can loose?
"Formally, this process reduces the dimension of the system from three to two" What is 3 and 2 here? The phase space is 4-dimensional. If the author means that an energy level should be fixed, he should say this explicitly.
"Since this system in not natural the question of nonlinear stability remains unanswered." First, "in not" probably means "is not". Secondly, such questions can be answered by using arguments from KAM-theory. Now this is quite standard.
I have only a few small comments that I would like to see addressed:
- In the 2dof section, you say : If is natural (positive definite kinetic energy) a Krein collision cannot occur. I believe you mean a Krein bifurcation cannot occur.
- The section "Axisymmetric systems" begins rather abruptly. You should state that you are looking at a single particle in a potential, so that the Hamiltonian is T+U. For example you could move (13) or similar equation to this section. Shouldn't th Stark example be a subsection of this section?
- Instead of the \mapto arrow (for example in your limits in the Microwave section) I think you want just the \to arrow.
- The "Lagrange-Dirichlet" theorem (as Marsden calls it) is much more general than what you say: If the Hessian of H is positive definite at an equilibrium, then it is stable. (No need for a natural system here). You should at least mention that you require a minimum of U. Is that what you mean by a "quadratic equilibrium"?
- I think you should add a reference to the text Meyer, K. R. and G. R. Hall (1992). Introduction to the Theory of Hamiltonian Systems. New York, Springer-Verlag.
Just to explain what is new. I agree with all of Reviewer B's comments and have already included #1,3,and 5. I will have to give # 2 and 4 some thought. I did previously define "natural flow" in the intro. I will shortly be including the Lagrange-Dirichlet theorem in its most general form. Whether Krein bifurcations can occur in higher dimensional natural flows is unclear and I would welcome readers' input on this point. I also took the opportunity to add "DEFINITION" in 3 places, just for clarity. I already had included the Meyer and Hall ref in the 2nd article and now it is also in the first, but so far with no citation in the text.
I have attempted to respond to all the referee's suggestions. The Lagrange-Dirichlet Thm is now included. Both the Stark and Microwave examples have been made subsections ot the Axisymmetric Systems section, where I remarked that H = T + U. A few very minor cosmetic changes have also been made. I plan to replace Fig. 4 with a better version in the near future. Thanks for the very useful comments.
Unfortunately I am not ready to approve the article because I do not see any reaction of the author to any my comment in the review. 05.04.2011 Reviewer A.
All of Reviewer A's comments are now incorporated into the article. In particular the title was changed to reflect the smaller scope.