chaosbook:conjug

# Chapter: Get straight

(ChaosBook.org blog, chapter Get straight) — Predrag Cvitanovic 2009-03-08

(the latest posts within a section at the top, blog-style)

Gregor, if you move helium section to chapter "Hamiltonian dynamics" why don't you also start editing/rewriting this chapter too — Predrag Cvitanovic 2009-03-29

This chapter doesn't really fit in nicely. It is split into three different parts with 6.1, 6.2 and 6.4, 6.5, 6.6 fitting together nicely and 6.3 "Collinear helium" sitting somewhat uncomfortably in the middle. — Gregor Tanner 2009-3-26

Re. 6.X (X.ne.3), I am not sure whether the rest of the material makes a full chapter and is at the right place - it is after all quite advanced material for somebody who just got his/her head round cycle stabilities. One could of course always send it into the abyss of becoming appendicised, but there is too much nice stuff in the chapter for that.

Material covered in this chapter is needed for the semi-classical helium chapter, and for desymmetrization chapters, where coordinate changes are one way to map an equivariant flow into an invariant flow. Currently does not fit into the flow of the ChaosBook. — Predrag Cvitanovic 2009-03-10

Nonlinear coordinate transformations are very important but we do not use them much in ChaosBook. This chapter collects in one place all that we will need. It is meant to be omitted on first reading, sections of it read later, when need arises.

Necessary stuff:

• cycle stabilities are metric invariants of flows
• turn a Keplerian ellipse into a harmonic oscillator
• regularize the 2-body Coulomb collisions in classical helium
• what else?

Rewrite the header of the chapter and the summary so the motivations (as listed in this blog) and the main results are written up and clear.

## Section: Changing coordinates

Sounds good - make this preamble of "Rectification of maps". — Predrag Cvitanovic 2009-03-29

I suggest integrating this section, “Rectification of flows” and “Rectification of maps” into one, currently 6.4 "Rectification of maps". After all linearizing the flow is the first order term - so one could introduce it as a generalization of the methods taught previously. — Gregor Tanner 2009-3-26

Collect clearly and concisely formulas needed later on, especially for linearized flows

## Section: Rectification of flows

OK, make this first subsection of "Rectification of maps". — Predrag Cvitanovic 2009-03-29

Harmonic oscillator change of coordinates needed later to motivate Rossler flow, and equivariant → invariant coordinate changes. Make into a subsection? — Predrag Cvitanovic 2009-03-10

## Section: Collinear helium

If you want to move it to the end of chapter "Hamiltonian dynamics", it is OK with me. I would place it ahead of Section 7.4 “Poincaré invariants,” which is of narrower interest. Originally it was there, but it did not make sense without defining smooth coordinate transformations first. If you harmonize it with this chapter, might work well. BTW, almost any cojugacy is singular somewhere. For example, if you try to flatten out a neighborhood of a fixed or periodic point, the conjugacy has at best a finite radius of convergence. — Predrag Cvitanovic 2009-03-29

While I see the idea behind trying to match KS regularization with smooth conjugacy, I am not sure it works well. Collinear helium is singular (and you want smooth flows to start with), the transformation involves time (which is a new concept at that point), and one of the main points of the KS transformation is that the combination of time and space transformation keeps everything Hamiltonian (a point which comes too early here). In addition, this text was written with the helium quantization chapter, so it needs to be rewritten further.

I would move this section to the end of chapter "Hamiltonian dynamics" and work out the connection to Hamiltonian dynamics and regularization/smooth conjugacies there. — Gregor Tanner 2009-3-26

regularize the 2-body Coulomb collisions in classical helium.

## Section: Rectification of maps

OK, make this 2nd subsection of "Rectification of maps". — Predrag Cvitanovic 2009-03-29

If material in Chapter Noise gets sufficiently developed, up to how to include weak noise perturbative corrections (as a warm-up for the corrections), this material will be needed. Otherwise, move into an appendix? — Predrag Cvitanovic 2009-03-10

## Section: Rectification of a periodic orbit

Not sure what you mean by “remark” - in ChaosBook.org they are scholarly pointers to history and literature relevant to the chapter, collected at the end of the chapter. Remarks within the text that state something but do not explain it do not belong to a textbook; if you cannot explain the material, best not to include it. Or make this 3rd subsection of "Rectification of maps". — Predrag Cvitanovic 2009-03-29

One can probably drop "6.5 Rectification of a periodic orbit" - or make it into a remark in 6.4 "Rectification of maps". — Gregor Tanner 2009-3-26

Perhaps make into a subsection of the preceding section? — Predrag Cvitanovic 2009-03-10

## Section: Missing material?

Is some important material missing from the book? In particular, this text has not yet been harmonized with the needs of Chapter "World in a mirror" and yet mostly unwritten (but essential) Chapter "Continuous symmetries".

## Section: Cycle Floquet multipliers are metric invariants

Originally it was there, but that does not work - cannot introduce smooth conjugacy without explaining what smooth conjugacy is - that is how this chapter was born. Remember, each chapter should be not bigger than what you can cover in 1 - 1 1/2 hours lecture. Try again. — Predrag Cvitanovic 2009-03-29

This section could become a part of 5.2 Floquet multipliers are invariant (with a chili pepper attached). — Gregor Tanner 2009-3-26

This is the most important section: cycle stabilities are indeed metric invariants of flows. The reason why this chapter was created.

## Rėsumė

Mark here when summary completed.

## Commentary

Here all references to external literature (avoid them in the text proper, it is meant to be self-contained).

## References

Mark here when correctly ordered. OK to have references not cited, as long as they are relevant to this chapter.

~~DISCUSSION~~

chaosbook/conjug.txt · Last modified: 2010/02/02 07:55 (external edit)