ещё о классической физике + принцип стабильности

[avva]

Взял в библиотеке две книги: "Математические методы классической механики" Арнольда и "Foundations of Mechanics" by Ralph Abraham (1967, с сильным уклоном в математическую сторону). Кроме того, нашёл сетевую версию книги Structure and Interpretation of Classical Mechanics, которая, кажется, как раз что надо. Когда я всё это буду читать, тем не менее, не имею ни малейшего понятия. Спасибо всем, кто рекомендовал учебники.

В предисловии к книге Абрахама обнаружился интересный отрывок о стабильности физических систем. Возьму-ка я и процитирую его полностью. Выглядит очень интересно, но не успел пока над этим подумать как следует.

At the turn of this century a simple description of physical theory evolved, especially among continental physicists -- Duhem, Poincare, Mach, Einstein, Hadamard, Hilbert -- which may still be quite close to the views of many mathematical physicists. This description -- most clearly enunciated by Duhem[1] -- consisted of an experimental domain, a mathematical model, and a conventional interpretation. The model, being a mathematical system, embodies the logic, or axiomatization, of the theory. The interpretation is an agreement connecting the parameters and therefore the conclusions of the model and the observables in the domain.

Traditionally, the philosopher-scientists judge the usefulness of the theory by the criterion of adequacy, that is, the verifiability of the predictions, of the quality of the agreement between the interpreted conclusions of the model and the data of the experimental domain. To this Duhem adds, in a brief example [1, pp. 138 ff.], the criterion of stability.

This criterion, suggested to him by the earliest results of qualitative mechanics (Hadamard), refers to the stability or continuity of the predictions, or their adequacy, when the model is slightly perturbed. The general applicability of this type of criterion has been suggested by Rene Thom [2].

This stability concerns variation of the model only, the interpretation and domain being fixed. Therefore, it concerns mainly the model, and is primarily a mathematical or logical question. It has been studied to some extent in a general logical setting by the physicologicians Bouligand[3] and Destouches[4], but probably it is safe to say that a clear enunciation of this criterion in the correct generality has not yet been made. Certainly all of the various notions of stability in qualitative mechanics and ordinary differential equations are special cases of this notion, including Laplace's problem of the stability of the solar system and structural stability, as well as Thom's stability of biological systems.

Also, although this criterion has not been discussed very explicitly by physicists, it has functioned as a tacit assumption, which may be called the dogma of stability. For example, in a model with differential equations, in which stability may mean structural stability, the model depends on parameters, namely the coefficients of the equation, each value of which corresponds to a different model. As these parameters can be determined only approximately, the theory is useful only if the equations are structurally stable [т.е. малые изменения параметров ведут к малым изменениям решений -- avva], which cannot be proved at present in many important cases. Probably the physicist must rely on faith at this point, analogous to the faith of a mathematician in the consistency of set theory [интересная аналогия! выделено мной -- avva].

An alternative to the dogma of stability has been offered by Thom[2]. He suggests that stability, precisely formulated in a specific theory, be added to the model as an additional hypothesis. This formalization, despite the risk of an inconsistent axiomatic system, reduces the criterion of stability to an aspect of the criterion of adequacy, and in addition may admit additional theorems or predictions in the model. As yet no implications of this axiom are known for celestial mechanics, but Thom has described some conclusions in his model for biological systems.

A careful statement of this notion of stability in the general context of physical theory and epistemology would be quite useful in technical applications of mechanics as well as in the formation of new qualitative theories in physics, biology, and the social sciences.

[1]: Duhem, P. The Aim and Structure or Physical Theory, Princeton Univ. Press, 1954.
[2]: Thom, R. Stabilité structurelle et morthogenèse, Benjamin, NY, 1967.

Интересно, что за биологические примеры описывает Том?

Если кому-то есть что добавить по этой теме, добавляйте, пожалуйста.

Comments

[centralasian.livejournal.com]

tom zanimalsya teoriey katastrof... i vsega rassamtrival mnogo primerov biologicheskih sistem - kletka, starenie organizma, dinamika ekosistem... plys, on razrabotal intersniy matematichesky apparat dlya analiza processiv izmeneniy v sistemah... system changes theory, or something...

po-moemu, est dazhe perevody na russkiy... no oh, kak vsyo eto zarzhavelo v moei golove :((

[hotgiraffe.livejournal.com]

Арнольд - хороший (учебник, в смысле).
Ещё мне, кажецца, нравились пара глав в "Современной геометрии" Фоменко сотоварищи, где излагаецца теормех (вариационное исчисление, Лагранж-Гамильтон).
Всё это было так давно ;)
До сих пор жалею, что почти не слушал в институте второй семестр теормеха, где его систематически рассказывали на основе непрерывных групп преобразований.

[averros.livejournal.com]

The notion of stability of biological systems is somewhat outdated (and easily demonstrated as false by numerous examples of sensitivity of cells to single hormone molecules, sometimes disastrous results of single nucleotide mutations (a single mutation in p53 gene in a single DNA molecule can cause a cancer killing an entire organism), or even sensitivity of rod cells to single photons).

The biological systems (and, increasingly, large technological systems, like the Internet) are fragile (i.e. can be catastrophically damaged by small events) but robust (i.e. contain diverse mechanisms for adjusting and reacting to the most common environmental influences or internal break-downs). See, for example http://www.its.caltech.edu/~leectr/workshop02/DoyleComplex.pdf and percolation theory.

The stability of classical mechanics is also somewhat suspicious; even in the absence of thermal noise it is possible to build simple, but unstable systems (for example, cellular automata). Also, it is well known by now that n-body problem (with n > 2) tends to have decidedly unstable solutions (i don't have a reference handy, unfortunately).

I think that the dogma of stability is pretty much dead; it is mostly an artifact of inability to analyze interestingly complex systems analytically, in effect disregarding unstable classical systems because there was no way to study them. Numerical modeling at discretization fine enough to convincingly separate effects of small changes in initial conditions from discretization error accumulation was simply unavailable in 60s.

Interestingly enough, noise tends to make systems to appear stable by quickly eliminating metastable states (an interesting problem facing designers of asynchronous logic elements is arbitrage between simultaneous signals - transistors being analog devices have non-saturated states; it turns out that "better quality" arbitration circuits are harder to get into metastable states, but then they spend more time deciding which way to go, thus degrading performance).

[malaya-zemlya.livejournal.com]

:Also, it is well known by now that n-body problem (with n > 2) tends to have decidedly unstable solutions (i don't have a reference handy, unfortunately).

Нестабильность в этой ситуации вызывается обычно столкновениями точечных частиц. Уравнения Ньютона в таких условиях бессильны что-то предсказать. Тем не менее, оказывается, что для n>4 можно соорудить нестабильную систему без столкновений.

В обзоре This Week's Finds in Mathematical Physics (Week 181) Джон Беез упоминает красивый пример нестабильности в задаче с 5 телами: в нем тела ухитряются разлететься на бесконечное расстояние за конечное время. Более подробно об этом написано здесь. Там же приводятся некоторые другие результаты теории стабильности задачи c n телами.

PS. В самом конце обзора Беез приводит ссылку на замечательную лекцию Арнольда озаглавленную On Teaching Mathematics. Очень советую прочитать.

[arish.livejournal.com]

Кажется, Том занимался биологией, когда уже не слишком узнавал своих близких...

Re:

[avva.livejournal.com]

Спасибо за ссылку на пример нестабильности. А Вашего мнения по поводу лекции Арнольда я не разделяю. Я уже читал её в прошлом, по-русски, и вследствие этого был даже жуткий флейм в ЖЖ.