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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 10th 2011
• (edited Jun 10th 2011)

I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.

For the time being, notice there was this old discussion box, which I am herby mving to the forum here:

+–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).

Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.

One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.

Toby: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word would you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)

Zoran: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.

Toby: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 10th 2011

to that end, I have started a reference-entry: Mathematical Topics Between Classical and Quantum Mechanics

to be expanded

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 10th 2011

okay, I have added some genuine (albeit basic) content: in new sections

• Classical mechanical systems

• Observables and states

• Flows and evolution

• Quantization

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 10th 2011
• (edited Jun 10th 2011)

I have also copied some paragraphs over to Poisson algebra.

I see that some $n$Lab entries emphasize that Poisson algebras are only about “finite dimensional” classical systems. I am not sure if that is a wise restriction to make in that generality. I’d rather vote against it. Also I’d vote against making too strong a distinction between “classical mechanics” and “classical field theory”, as some entries seem to do. It’s better to emphasize how there is a big picture that subsumes this all, than to reinforce superficial distinctions. But have to run now…

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeJun 11th 2011

We already have the page classical physics. As the discussion with Zoran in #1 attests, I don’t know whether the distinctive meaning of ’mechanics’ that I was taught –that mechanics is about particles, not fields– is correct, or if the distinction that Zoran was taught –apparetnly, that mechanics is about systems that are (classically) deterministic on a fundamental level, not thermal–, but if that word doesn’t have some content, then why aren’t we just putting all of this stuff at classical physics?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 11th 2011
• (edited Jun 11th 2011)

I think it’s true that the standard textbooks mosty take “mechanics” to refer to refer to finitely many degrees of freedom. But I don’t think that’s a particularly helpful point of view, and I’d rather de-emphasize it on the $n$Lab.

I’d rather go the route that I have now started implementing: say a bare bones “classical mechanical system” is a real Poisson algebra, and a bare-bones “quantum mechanical system” is a complex star-algebra, and then iteratively add structure as desired (for instance finite manifold structure for single point particle dynamics, or Fréchet manifold structure for certain field theory dynamics, etc.).

Concerning the distinction between classical physics and classical mechanics. Not sure. Currently the former does mention some things that would not really fit into the latter, such as “classical optics”, so I guess it makes sense to have two entries.

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeJun 12th 2011

I don’t get it. If Maxwell’s equations are mechanics, then why not classical optics? (The latter even has finitely many degrees of freedom.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

I think “classical optics” is not quite the same as the study of the classical Maxwell field. This is about refraction, lense forumulas, etc, which require further idealization and assumption.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011

Urs, when you remove the discussion from an entry you should leave there a visible permanent link to the archiving place!

Toby, I do not know if you quote here me:

mechanics is about systems that are (classically) deterministic on a fundamental level,

but this sentence as stated does not make sense. Any theoretical description is a theoretical description – it has variables and so on. Like in logic. It does not make sense to say that it is such and such “at fundamental level”. A desription contains those variables which it contains. Therefore, thermal physics does not know that it can be derived as a different hierarchical level of a description of mechanics, namely in thermodynamics limit of descriptions in statistical physics of certain (say microcanonical) ensembles in the usual mechanical systems. Thus there is a completely different framework of thermal physics which is believed in principle to be derivable from a statistics approach (however the counting of states and some other principles in statistical physics are hard not to take at least some amount of quantum principles, even in limit $h\to 0$).

Urs, I do not understand the insistance that “standard textbooks” take mechanics by definition to have finitely many degrees of freedom. If you mean second-level undergraduate textbooks, I do not know as I do not read this trash. Mechanics of continua, mechanics of fields, mechanics of vibrating rods, classical mechanics of plates and membranes and so on, are abundant in journals and in what I call standard textbooks, that is Landau, Goldstein, works of applied mathematician Marsdein and so on.

Urs, I like more your approach in 6, but I think one should not be reductionist and assume everything can be covered by one definition. This is a bit of marginal but physicists sometimes cover those as classical. There are systems which have lagrangean but the passage to supposed hamiltonian has a degenerate matrix so hamiltonian does not exist (an exrecise in Goldstein, in my memory). There are effective classical systems which are not Newtonian, hence correspond to variational systems where higher derivatives are in place, thus do not belong to Poisson mechanics. Poisson mechanics is just one dominant approach, which does cover Newtonian systems what is classical mechanics in narrow sense.

As far as classical optics, it is the short wavelength limit of wave equations. Formally it is analogous to classical mechanics, it is the same eikonal limit in which Schroedinger wave equation solution passes to classical mechanics (WKB approximation). Maxwell equations are classical equations of a system of fields, classical mechanics of fields, WKB approximation is creating a different system which can not be described fully in that sense. There is the least path functional coming from it, which can be described by a Lagrangean of a sort, however, the things like the phase shifts in focal points and so on *behaviour of caustics) which follows from WKB can not be explained by those.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

Classical statistical mechanics is not, strictly speaking, part of classical mechanics, as it studies a different entity: the statistical ensemble; of course the ensemble consists of a collection of classical mechanical systems, but this is the same as saying that the group theory for example is not part of a set theory as a subject. Of course, groups are usually described by means of sets with a structure. Thermal physics is hoped to appear in the limit of statistical mechanics ensembles, but this is proved only for a limited number of phenomena and usually in special cases of ensembles of systems of rather special type. It is just a conjecture that it may be obtained that way in full generality.

There are many things which do not belong to classical mechanics and are in classical physics.

Toby, 7: it is not true that classical/geometrical optics is a system with finitely many degrees of freedom. Some of the hi school exercises are of such nature, but this is not true for geometrical optics in general.

Toby, 1: “any more than you have a slick phrase for ‹mechanics of point particles”

Toby, this is not a slick phrase but a standard terminology and the name of chapters in major monographs, like Hatfield, Goldstein and so on. This is true for classical mechanics as well as for quantum mechanics.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

I think if one defines classical mechanics in a single formalism, then it would be better to have it in a page on that formalism. I mean classical mechanics is entry page about the subject. If one defines the Poisson formalism, then maybe it woudl be better to place it in a separate entry Poisson mechanics. In classical mechanics textbooks it is a chapter, hence in $n$Lab it should be an entry. Do we agree ? I can move it later.

But, the underlying algebra of a Poisson algebra is by the definition commutative (or graded commutative for superPoisson algebra), especially in classical mechanics context. One can consider the noncommutative notion by emphasising it in the title (like the rare notion of “nonassociative ring”).

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

The Landsman’s book which you are using (and also in the encyclopaedia article linked at Poisson algebra) is also definite about the commutativity of the underlying product (for Poisson algebra), def. 1.1.2, page 37/38 (the danger is that most of the standard theorems on Poisson algebras do not hold for the generalization where the ordinary product is not commutative). The noncommutative generalization is extremely rare and nonconventional, for example in noncommutative symplectic geometry of Kontsevich. This would be a hypothetical subject of a “classical mechanics on a noncommutative space”. Edit: I added some references to Poisson algebra and made changes and a remark to the definition.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJun 12th 2011

Hey Zora, I did consider commutative Poisson algebras.

I am not sure which more general classical mechanics you want to consider that is not based on Poisson algebras.

• CommentRowNumber14.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

E. g. multisymplectic manifolds, Nambu mechanics and those Lagrangean systems which have Lagrangean but the conditions for the Lagrangean transform to the Hamiltonian are not satisfied.

P.S. The very word “Poisson algebra” standardly includes that the ordinary product is commutative. Zora is a female name; one of my aunts is Zora. :) My main argument against having defining classical mechanics as a Poisson algebra is not only because some other cases are sporadically used, but in the fact that classical mechanics is a subject regardless the formalism and Poisson mechanics is a particular formalism used. The subject of music is not a particular theory of music. So, to keep it useful for the end user, the entry on Poisson mechanics should contain the Poisson formalism, and on classical mechanics general ideas and list of formalisms like Newtonian, Lagrangean, Hamiltonian, Nambu etc.

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

Finally, why do you take as the category of classical mechanics the opposite of the category of all commutative Poisson algebras over reals, and not the smaller but more correct category of Poisson manifolds ? I admit that it is interesting to consider that generality for studying various generalizations and as an input for quantization, the classical mechanics in the standard sense does not allow more general Poisson algebras than (algebras of functions on) smooth (possibly infinite-dimensional) Poisson manifolds. I mean, the general Poisson algebra may have nilpotent elements (in the commutative algebra part) what does not belong to the setting of classical mechanics. Do you want to live with nilpotents (by your definition it is even an observable then!) and with non-smooth spaces ?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJun 12th 2011

The bare-bones structure needed for the standard notions of classical mechanics is that of a Poisson algebra. Next one can add refinements to this: smooth manifold structure, scheme structure, infinite-dimensional manifold structure, diffeological structure, etc. If I have time I’ll add more on such refinements. Unless you do it first.

Please feel free to remove the mention of non-commutative Poisson algebras, if you think it is a problem.

• CommentRowNumber17.
• CommentAuthorzskoda
• CommentTimeJun 12th 2011
• (edited Jun 12th 2011)

That is not representative of classical mechanics. These are mathematical fictions motivated by and generalizing the classical mechanics in one particular direction (variational calculus with higher derivatives does it in a different direction, dynamical systems with discrete time in third direction and so on). Classical mechanics is part of the physics and Bourbaki minimalistic generalizations are interesting but should be separated from the description of the subject. I do not see how can a nilpotent element of an algebra be considered an observable, how does one measure something what squares is zero for example. I can not be happy contributing to a page which makes a false impression of the subject, and whose intention is unclear. It would be more correct to displace that material to a separate page, say classical mechanics of Poisson algebra where one can look at that particular generalization. By standard notion of classical mechanics I would not think just of those listed but of many others which hardly generalize to such generality (like Legendre transform, Poisson leaves, canonical transformation and so on).

• CommentRowNumber18.
• CommentAuthorGuest
• CommentTimeJun 13th 2011

I would certainly agree that there are systems considered in classical mechanics that do not have a Poisson structure. Dissipative systems are a main example.

I would also agree that the word "mechanics" does not imply finitely many degrees of freedom, and that continuum field theories are special cases of classical mechanical systems. Classical treatises on the subject, like Goldstein, also agree, I believe.

On the other hand, higher derivative variational principles do induce natural symplectic and Poisson structures. This follows from the old work of Ostrogradski and its modern descendent the covariant phase space method. The same method implies that degenerate Lagrangians are invariant under gauge transformations and that natural symplectic and Poisson structures appear after quotienting by gauge transformations. Arguably, only the quotient should be considered as a true mechanical system. Even variational principles with discrete time steps may be given a symplectic (and hence a Poisson) structure, cf. Marsden's discrete mechanics.

Non-commutative algebras certainly do appear in the classical limit of quantum theories with fermions, though they are supercommutative and the anti-symmetry and Jacobi axioms of the Poisson bracket are modified accordingly. It is possible that even richer non-commutative structures are possible, which may arise in the classical limit of something like anyons. However, such exotic examples are rarely considered and are outside my experience.

The cost of extending classical mechanics to classical statistical mechanics of ensembles (I'm excluding Boltzmann's kinetic theory, which is dissipative) comes at almost no cost (state as homomorphism vs state as positive linear functional). I see no reason why not include it.

I'm not completely sure why "mechanical systems" should be identified with the opposite category of Poisson algebras. In principle, one could draw the distinction between Poisson algebras and their underlying Poisson manifolds, which is parallel to "mechanical systems" and their "phase spaces". The two categories are essentially opposites of each other, but still have different names.

Igor

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

I have added to classical mechanics

• the remark that generally the Poisson algebra in qustion is a super-Poisson algebra for systems with fermions;

• the remark that some variants of “mechanical systems” may not be modeled by a Poisson algebra, such as dissipative systems, which are however to be regarded as approximations to larger systems that are described by a Poisson algebra.

Igor, I don’t understand your last remark, where you say

The two categories are essentially opposites of each other, but still have different names.

That seems to be exactly what the entry says. What are you suggesting to change?

• CommentRowNumber20.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

Thank you Igor for the physical perspective, you talk my (more conservative) language here :) In the idea section we indeed stated before that the Poisson structure is for non-dissipative systems. By slick devices like Feynman-Vernon Ansatz one can however introduce additional sinks and sources to make a double dimensional non-dissipative system out of dissipative. I do not remember the details, though I have seen of higher derivative variational calculus reducing to the usual, but I suppose this is also happening only by enlarging the dimension of the space by introducing new variables, no ?

States of a mechanical system and a state of an ensemble are physically quite different entities. I understand that both in classical and in quantum mechanics the difference between pure and mixed state brings up the difference between a configuration of the system and a statistical mixture. This is useful in classical mechanics when treating things like Louville’s theorem.

I however agree with Igor that Poisson algebras should be viewed as opposite to the system. I disagree slightly about calling classical mechanics for completely arbitrary Poisson algebra – the nilpotents certainly do not appear in what is usually viewed as classical mechanics, though have nothing against it in a theory, just I think it does not belong to the intro entry on what classical mechanics is.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

the remark that some variants of “mechanical systems” may not be modeled by a Poisson algebra, such as dissipative systems, which are however to be regarded as approximations to larger systems that are described by a Poisson algebra

If one looks at the whole world at once, then one is eventually coming to the unification of everything, hence one has quantum mechanics eventually. Many dissipative processes need a molecular dynamics for an explanation, so one gets into microscopic theory, which includes heat and quantum mechanical processes, incuding chemical processes. So it is not true that purely mechanical picture in the sense of Poisson algebra can explain dissipative processes in general. So we get outside of mechanics, and as long as we want to stay within mechanics we have to accept that there exist open systems without conservation of energy and we can not expect that the conservation of energy has to be satsfied within the realm of classical mechanics. Yes, in quantum physics but not in classical mechanics a la Poisson algebras.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

I however agree with Igor that Poisson algebras should be viewed as opposite to the system.

I still don’t understand what you mean to imply by this: that’s what the entry says.

• CommentRowNumber23.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

Sorry, it is a typo!!! The sentence was intended to be

I however agree with Urs that Poisson algebras should be viewed as opposite to the system.

I added a new paragraph to the entry classical mechanics on open vs. closed systems.

• CommentRowNumber24.
• CommentAuthorGuest
• CommentTimeJun 13th 2011
"Igor, I don't understand your last remark, where you say

"The two categories are essentially opposites of each other, but still have different names."

That seems to be exactly what the entry says. What are you suggesting to change?"

Sorry, Urs. I guess it was not completely clear. My first reaction would be to use "classical mechanical systems" as a synonym for CPoiss. If parallel synonym for CPoiss^op would be needed, I'd use "classical phase spaces". I was wondering if there was a reason to exchange the roles of CPoiss and CPoiss^op.

I am somewhat agnostic as to whether all Poisson algebras could be considered mechanical system, or only some subset of them.
• CommentRowNumber25.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011
• (edited Jun 13th 2011)

Igor, the points of phase space of a mechanical subsystem are the points of a subspace of a phase space. Hence the subobjects are in the direction of inclusion of spaces. If one would take $\mathrm{CPoiss}$ for the true category then the phase subspace would be a quotient object in $\mathrm{CPoiss}$. And so on.

Do you really want the Poisson leaves, for example, to be quotient objects in your category of mechanical systems ??

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

Okay, I see. So the careful statement should be to just say what the category of classical systems is. Everybody is then free to present this category in any way using any objects he pleases.

So the entry says: the category of classical mechanical systems should be the opposite of that of (super)Poisson algebras. I guess that’s uncontroversial?

• CommentRowNumber27.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

I think that an intuitive physical meaning of a physical subsystem corresponds to both the configuration and phase subspaces, not the opposite, hence I still agree with Urs’s convention. It also agrees with the Gelfand-Neimark theory taking characters/pure states as geometric points.

• CommentRowNumber28.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011
• (edited Jun 13th 2011)

Urs, what is the meaning of the derivative in the section on Flows ? In your generality $A$ has no topology, it is just a vector space, so I do not see how can one differentiate a one parameter family of linear maps $A\to A$ with respect to a parameter.

P.S. I changed the thing about inclusion into bigger “fundamental system” into more accurate

This definition captures most notions of “mechanical systems”. Exceptions include open systems for which there is no conservation laws (examples: externally-driven and dissipative systems). In real worlds, physicists believe that such systems may be realized only as parts of larger systems (eventually, the universe) which are conservative; hence either describable by a Poisson algebra or it entails energy types which can not be described using classical mechanics.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

Right, so at that point I was maybe thinking of a finite dimensional vector space. That needs to be improved on.

• CommentRowNumber30.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011
• (edited Jun 13th 2011)

Poisson algebra of observables is almost never finite-dimensional in practice.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

Yes, but I was thinking of a finite dimensional vector space. Don’t have time for it now. If you have, please add the relevant extra clauses. Or a remark that leaves it to the reader…

• CommentRowNumber32.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

I created a related stub dynamical system.

• CommentRowNumber33.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

I made a quick fix to flow, it should be examined if it is sensible. I travel to Zagreb in a bit.

• CommentRowNumber34.
• CommentAuthorTobyBartels
• CommentTimeJun 13th 2011

@ Urs #8: Yes, that’s how I interpreted it [classical optics]. Why isn’t that mechanics? It’s an extremely simplified special case of Maxwell’s equations, which apparently is mechanics, so why isn’t that?

• CommentRowNumber35.
• CommentAuthorTobyBartels
• CommentTimeJun 13th 2011

@ Zoran passim: I still don’t understand what you mean by “mechanics”. It’s not about being deterministic, which was just a guess from your exclusion of thermodynamics, OK. It’s not about what the people who study it call “mechanics”, because “statistical mechanics” is not mechanics. It’s not about having a Lagrangian or Hamiltonian or Poisson algebra, because these are specific formalisms that don’t include every mechanical system (with which I agree). It’s certainly not the “second-level undergraduate” meaning that I learnt; I can accept that.

So what the heck does it mean???

• CommentRowNumber36.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

Toby 34 on geometrical optics – You are right, the eikonal equation looks like it could be viewed as a classical field theory, but as such it is not a uniquely defined system. It is an asymptotic theory which suffers for example Stokes phenomenon – the asymptotic expansions are well defined only in certain sections of the space, and there are discontinuities (described by Stokes factors) when passing from a region to region. The Maxwell Equations are on the other hand linear, and as for the linear wave equations one does not have such phenomena; moreover the mechanics of continua is usually derived as a limit of a discrete model when the spacing goes to zero, what makes it close to discrete mechanical systems.

Toby 34 on statistical mechanics – Mechanics describes both point particles and extensive objects of matter. Statistical mechanics does not describe single instance of a matter but an abstract entity, a statistical ensemble of systems which have some physical parameters in common, but otherwise vary enormously in detail.

I agree with you that it is hard, if possible, to give an abstract definition of mechanics without a recourse to formalism. But the subject is defined rather historically, by studying classical mechanics of point particles and extensive bodies defined only in terms of basic (not specific for a kind of matter) kinematical and dynamical quantities like force/pressure, energy, mass, density. The further features like magnetization, charge etc. possessed by bulk are just ways of expressing the contributions to energy, but they require additional dimensions in the game. This is the point where one can usually stays within formally the same formalism but the same kind of laws are applicable. In addition to mass density one has charge density, magnetization density and so on. Well mas is nothing special it is an effective quantity anyway. There are differences between various discrete and continuum formalisms but one can be obtained from other in regular cases or by limiting procedure, so they are all interconnected. In hydrodynamics one has already a problem: while formally one can proceed the same way, physically, for compressible liquids, the thermodynamic variations are of importance.

So I agree, there are problems, but historically some similarities and correspondences between good cases of formalisms define subgroups of physical theories and some ready differences like between matter systems and statistical ensembles are taken as imortant in defining the standard boundaries of the field. I do not see a solution to your objections (and mine) but see rather, to some precision what the features of the historical groupings are.

• CommentRowNumber37.
• CommentAuthorzskoda
• CommentTimeJun 13th 2011

I should also note that the formal similarities between mechanics and thermodynamics hold only for equilibrium thermodynamics. Non-equilibrium thermodynamics is quite different. Fundamental mechanics, statistical mechanics and thermodynamics are often viewed as different hierarchical levels of describing multiparticle systems. Their distinction is over all physical.

• CommentRowNumber38.
• CommentAuthorTobyBartels
• CommentTimeJun 13th 2011

I really don’t understand this claim that statistical mechanics is not about matter!

Classical statistical mechanics is a method for studying what appears (in the ignorance of quantum phenomena) to be the real world: a classical physical system made up a large number of point particles (and perhaps other things). In other words, it is about matter. It is not about statistical ensembles any more than classical mechanics is about Poisson manifolds. Although its methods are specialised, fundamentally it is simply the application of probability theory to deterministic physics.

Furthermore, all of physics is about matter (and energy/radiation, which is not fundamentally different). If it’s not about that, then it’s not physics!

So I remain in ignorance of the difference between mechanics and physics.

• CommentRowNumber39.
• CommentAuthorTobyBartels
• CommentTimeJun 13th 2011

Certainly there are “standard boundaries” and “historical groupings” that distinguish mechanics per se from the more general subject of physics. However, there are several competing ideas of what this distinction is! You learnt one, Urs learnt another, and I learnt yet a third. (The way that I learnt it, classical statistical physics, when applied to point particles and rigid bodies, is mechanics, but continuous phenomena are not. You learnt about the opposite. Urs seems to have learnt the most restrictive definition, although he wants to reject it in favour of yours, or something like that. Igor seems to advocate the most inclusive version, in which continuous and statistical physics are both mechanics.) Of course all of the differences between these fields are real, but where to draw the line is starting to seem very arbitrary.

I would like to see clear evidence that there is a consistent usage of terminology in modern physics, but right now I don’t believe that. If there is such a consistent terminology, then we may as well have a page on it. If, however, there is not, then we should have (in addition to classical physics) pages on specific ideas (such as physics that can be described using Poisson algebras, which is what classical mechanics is about right now). I suppose that we might still have a page linking to fields of classical physics that are called “mechanics” (including continuum and statistical mechanics), but we could just as easily put that (along with the other links) at classical physics.

For what it’s worth, Wikipedia mostly agrees with me; it includes continuum mechanics (such as hydrodynamics) but distinguishes this from field theory (which is not included), and it includes classical statistical mechanics. However, that may just confirm that my education in terminology stopped as a second-level undergraduate.

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeJun 13th 2011

Hey Toby,

not sure how to resolve all this right now. If you feel the entries need to be re-arranged, I suppose I’ll be fine with it all.

What I’d like to see most is more genuine content added to all the entries in question.

• CommentRowNumber41.
• CommentAuthorzskoda
• CommentTimeJun 14th 2011
• (edited Jun 14th 2011)

Toby, of course, statistical mechanics applies to physics, but the content of its models is formally not a concrete matter system, but an abstract ensemble. Statistical physics in practice depends also on many principles which are not provable from pure statistics of mechanical models in general (and most often not true in full generality) like detailed balance and ergodic hypothesis. These principles are getting more clarified in modern, more rigorous research in mathematical statistical mechanics, which is not accounted in these subtleties in most pratical applications of statistical mechanics. I have nothing against considering statistical mechanics part of mechanics, and it is sometimes considered that way, but as a subject it has enough peculiarities to be considered a different BRANCH of physics. What is far less consistent is your distinction between continuum mechanics and a field theory. There is no difference: continuum mechanics is a pretty standard example of a classical field theory. Stress-energy tensor, density (also things like magnetization in more elaborate examples) etc. of the matter in continuum are classical fields per excellence just like magnetic field strength, electric field strength in another example of Maxwell theory. Wikipedia usually tends to express the viewpoint of undergraduate textbooks, as far as physics is concerned, which exclude infinite systems from mechanics textbooks, except for (systems of) rigid bodies where there is a reduction of degrees of freedom due constraints.

Traditionally classical physics is not equating with classical mechanics as a subject. Of course, mechanics as a basic subject for all physics, through which variants, limits and slight generalizations in or out of proposed scope, is relevant for all the physics. I agree that at the fundamental level all physics is (quantum) mechanics, at least with what we know nowdays.

Urs: you could see that my explanations in previous entries HAVE a lot of content on aspects of variables and approximation used by various parts of physics. But I can not incorporate that in formal viewpoint which you obviously want (like mechanical systems is Poisson algebras). When I created entries classical mechanics and classical physics I was hoping to put some real physics inside. My education in physics is in condensed matter physics and similar subjects and not abstract novelties like string theory and sigma models where I am mostly illiterate.

• CommentRowNumber42.
• CommentAuthorzskoda
• CommentTimeJun 14th 2011

Urs and Toby: when I created these pages, I had in view, which I still back somewhat that the entries classical mechanics and classical physics are more of disambiguation pages or informal toc clusters with explanation for a future expansion of $n$Lab into the physics side; and that as widely used branch names should attract an outsider to the $n$Lab and direct toward what is our content we can provide and its relation to the outside world. That is why I prefer that particular formalisms be in depth developed on pages devoted to particular formalisms.

• CommentRowNumber43.
• CommentAuthorzskoda
• CommentTimeJun 14th 2011
• (edited Jun 14th 2011)

The way that I learnt it, classical statistical physics, when applied to point particles and rigid bodies, is mechanics, but continuous phenomena are not. You learnt about the opposite.

No Toby, I agree that foundations of statistical mechanics are mechanics. It is just a subject which developed (due its specific differences from the rest) as an independent branch. You know that usual thing that chemistry is part of physics. But still if we extend $n$Lab to chemistry, I think that chemistry should then have a separate entry! Statistical mechanics is in the same way a part of mechanics as chemistry of physics. I have no problem with putting statistical mechanics into mechanics in that sense and I did learn it that way. A newcomer looking for statistical physics and its thermodynamical limit will likely not look under classical mechanics (the latter should have a link somewhere, and I think it has to statistical mechanics and classical statistical mechanics eventually).

When I said that statistical mechanics is not about a matter (I meant many body system) but ensemble my intention was to show that it is strictly speaking formally about a different object than say Hamiltonian mechanics is. I do not say that the difference is very big, but that without defining a new object of study (ensemble) and, in practice, taking new level of independent assumptions, we do not get into the subject. Btw, the expression “statistical mechanics” is not yet involved in the entry classical mechanics so this is yet not critcial when talking the content of that entry.

But excluding continuua like elasticity and vibrating rods from mechanics. Even the engineer mechanics books do not do that and every physics introduction to this starts with a discrete system and INFERs (that is takes as a special case) what happens when the spacing becomes very small (you can call it limit, but physicist may alternatively think of a very small distance between the actual particles).

• CommentRowNumber44.
• CommentAuthorTobyBartels
• CommentTimeJun 14th 2011

@ Zoran: “What is far less consistent is your distinction between continuum mechanics and a field theory.”

That is not my distinction; that is Wikipedia’s. (I probably never should have brought that up; it’s just one place that I went to see if they’d worked out a consistent definition, and they really haven’t, although they seem to think that they have.)

At the moment, having seen that the distinction that I learnt is not generally accepted, I see no distinction between mechanics and physics. All of the various ways to specify mechanics as merely one branch of physics have proved arbitrary and not generally accepted.

• CommentRowNumber45.
• CommentAuthorTobyBartels
• CommentTimeJun 14th 2011

After reading #43, Zoran, I’m not sure that we disagree any more. You keep arguing against positions that I’ve abandoned, so I won’t repeat that I’ve abandoned them. But in writing ‘Statistical mechanics is in the same way a part of mechanics as chemistry of physics. I have no problem with putting statistical mechanics into mechanics in that sense and I did learn it that way.’, you also seem to be abandoning any attempt to say that statistical mechanics is not part of mechanics. So we are basically left with mechanics = physics.

In #41, you do write ‘Traditionally classical physics is not equating with classical mechanics as a subject.’. However, I think that we’ve proved this false. Or rather, there are several traditions, none of which agree. I suppose that everybody (except maybe Igor?) will agree that chemistry is not mechanics (although I would not now attempt to defend that), but everything else that’s been brought up has been considered mechanics by some traditions. So while some fields (such as statistical mechanics and continuum mechanics) have ‘mechanics’ in their names and others (such as thermodynamics and field theory) don’t, no other consistent distinction can be made.

I also agree with you that mechanics shouldn’t be identified with a particular formalism. (Incidentally, I was taught statistical mechanics through the Hamiltonian formalism! Of course, more was added on top of that.)

I like the idea of making classical mechanics and classical physics into overview pages that link to specific branches and formalisms, each on its own page. I just think that the former should redirect to the latter. I suppose that an alternative would be to discuss the various uses of the word ‘mechanics’ and what fields have been thought to be part or not part of classical mechanics, but the list of links on both pages would be duplicated, so we might as well have that discussion at classical physics.

• CommentRowNumber46.
• CommentAuthorzskoda
• CommentTimeJun 15th 2011

Toby, we do not know if all physics can be reduced to mechanics, but the contemporary opinion is that quantum mechanics together with theory of measurement does comprise all the physics. Quantum mechanics without the theory of measurement has a problem. You know that the wave function has a collapse/reduction in the process of measurement. Nobody know how that happens, is this instanteneous etc. There are interpretations, like Copenhagen says that these questions are out of realm of physics, but not all agree. In any case, the Schroedinger equation, that is the formalism of QM does not explain such things.

On the other hand, while I did agree that statistical mechanics can be taken part of mechanics, it is less true that the classical statistical mechanics can be taken part of classical mechanics in the full strict sense. Usually one starts statistical mechanics (say Feynman’s book or Landau’s book) with quantum picture of counting states, and the classical mechanics is just a case when the general Gibbs distribution is in a limit of Boltzmann distribution. I am not saying that it is impossible to do without quantum picture but even Boltzmann statistics is normally introduced with quantum picture in mind.

I think I can resolve things in the entry once. But I can not be driven by the momentary activities of others, being too busy with different subjects and duties now to pay enough attention to present burst of activities about mechanics pages.

• CommentRowNumber47.
• CommentAuthorzskoda
• CommentTimeJun 15th 2011

Toby, we do not know if all physics can be reduced to mechanics, but the contemporary opinion is that quantum mechanics together with theory of measurement does comprise all the physics. Quantum mechanics without the theory of measurement has a problem. You know that the wave function has a collapse/reduction in the process of measurement. Nobody know how that happens, is this instanteneous etc. There are interpretations, like Copenhagen says that these questions are out of realm of physics, but not all agree. In any case, the Schroedinger equation, that is the formalism of QM does not explain such things.

On the other hand, while I did agree that statistical mechanics can be taken part of mechanics, it is less true that the classical statistical mechanics can be taken part of classical mechanics in the full strict sense. Usually one starts statistical mechanics (say Feynman’s book or Landau’s book) with quantum picture of counting states, and the classical mechanics is just a case when the general Gibbs distribution is in a limit of Boltzmann distribution. I am not saying that it is impossible to do without quantum picture but even Boltzmann statistics is normally introduced with quantum picture in mind.

I think I can resolve things in the entry once. But I can not be driven by the momentary activities of others, being too busy with different subjects and duties now to pay enough attention to present burst of activities about mechanics pages. Despite agreement with yuo on principal issues I think it is opportune for the $n$Lab to keep different content at classical physics and classical mechanics. Also physics is usually including phenomena, theory and experiment; and by mechanics one only means the mathematical laws of physics (or engineering mechanics which is practical subject which is much more limited). If one wants to have any depth about mechanics in traditional sense in classical mechanics it would not be opportune to have much on thermal physics and so on there. On the other hand, I think one should have less specific of theoretical mechanics on classical physics page.

• CommentRowNumber48.
• CommentAuthorTobyBartels
• CommentTimeJun 17th 2011

Zoran, you write:

Toby, we do not know if all physics can be reduced to mechanics

At this point I have no idea whatsoever what this would mean!

When I say physics = mechanics, it is not to suggest that all physics can be reduced to mechanics, but rather that all physics already is mechanics, since nobody has suggested any other meaning of the term ‘mechanics’ (that hasn’t been shown to conflict with established usage).

Later, you speak of ‘mechanics in [the] traditional sense’. Even if this traditional sense is indefensible both theoretically (because it excludes field theories, perhaps) or linguistically (because it excludes things called ‘mechanics’, perhaps), can you explain what this is?

• CommentRowNumber49.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011

A delimitation of a subject is a conventional list of topics covered. As such it does not need to be defended, but remembered. When supplied with a list you object. But most of people orient by such lists including many potential users. For mechanics as for algebra there are several meaningful generalities possible. Neither of them is by the definition “all physical reality”, but some theories tries to evolve from starting with mechanics of point particles and taking various generalizations toward quantum fields and so on to believe that at the end that process of the generalization does include all physical laws. This is a meaningful thing, as it can be something of completely different nature complementary to that line of generalization.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeJun 17th 2011
• (edited Jun 17th 2011)

I have not been following this discussion. I wish similar energy would be invested into getting genuine content into the $n$Lab.

But if the question is still whether the entries classical physics and classical mechanics should be separate:

I think they should. The former lists stuff like thermodynamics which should not be the direct content of an entry on mechanics.

(Incidentally, much of thermodynamics has not even be fully rigorously been derived from mechanics. I once heard a talk by Jürg Fröhlich who had worked on the mathematical foundations of thermodynamics, and I was surprised to hear how comparatively little has been turned into theorems, and how comparatively hard that has already been. )

• CommentRowNumber51.
• CommentAuthorTobyBartels
• CommentTimeJun 17th 2011

When supplied with a list you object.

No, I have not been supplied with a list.

I supplied a definition that I learnt; you and Urs objected. Urs gave a definition; you objected. I have been told certain things about the list to which I objected, such as that statistical mechanics and classical optics don’t belong on the list, while field theory does.

But where is the list?

Here is a new idea, which I kind of like:

some theories tries to evolve from starting with mechanics of point particles and taking various generalizations toward quantum fields and so on

So perhaps mechanics is mechanics in the narrowest sense —the mechanics of point particles— and various generalisations thereof? So potentially all of physics, but only to the extent that it is a generalistion of the mechanics of point particles? Then membership on the list would be a matter of degree, with neither statistical mechanics nor field theory being mechanics par excellence, but both close and high up on the list?

I could buy that, but is that what you’re saying?

• CommentRowNumber52.
• CommentAuthorTobyBartels
• CommentTimeJun 17th 2011

There is a real question here, which ought to be easy to answer but has not been: what is mechanics?

I thought once that I knew the answer, but I was wrong. I looked up the answer on Wikipedia, and that was wrong (and also inconsistent). Nobody else has given an answer.

• CommentRowNumber53.
• CommentAuthorUrs
• CommentTimeJun 17th 2011

what is mechanics?

Oh, I see. That’s interesting. It wasn’t clear to me that this is now the question (I wasn’t following your exchange with Zoran).

I’ll try to come back to this a little later.

• CommentRowNumber54.
• CommentAuthorzskoda
• CommentTimeJun 17th 2011

So perhaps mechanics is mechanics in the narrowest sense —the mechanics of point particles

I do not know which of the historical figures in mechanics – Archimedes, Newton, Lagrange etc. would put point particles separate at least from rigid bodies, vibrating rods and few other simple mechanical continua. But logically this is true nowdays: we can start with discrete, even finite discrete system of point particles and go on with abstracting, taking limiting procedures and so on.