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Definition of "Classical Mechanics"


pmb
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I was wondering if anyone here has heard of the definition of Classical Mechanics which is defined in the well known, staple of classical mechanics Classical Mechanics - Third Edition by Goldstein, Safko and Pool, Pearson Education, Inc,. Publishing as Adison Wesley (2002). See page1, first paragraph

In the present century the term "classical mechanics" has come into wide use to denote this branch of physics in contradistinction to the newer theories, especially quantum mechanics. We shall follow this usage, interpreting the name to include the type of mechanics arising out of the special theory of relativity.

I have, for the longest time, used this text, and as such this definition for classical mechanics. So if you see me use the term you can be certain that this is what I mean. But I certainly can't assume that this is what others mean by it.

 

I'd like to make a request of you good folks to state the definition of classical mechanics as you choose to use the term, especially when knowing the precise definition ot it is critical.

 

Thank you all.

 

Pete

 

ps - If we were to take Goldstein etc. literally then the relativity forum would have to be absorbed into the Classical Physics forum. I think that'd be getting carried though. :P

Edited by pmb
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So what's wrong with absorbing relativity into 'classical'?

 

Where would you place relativity and quantum mechanics?

 

My distinction would be between the mechanics of continua (which includes relativity) and the mechanics of discrete systems (which includes quantum mechanics ) and also scale dependent mechanics.

Edited by studiot
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So what's wrong with absorbing relativity into 'classical'?

I don't see anything wrong with it in principle. In practice, in this website it might just be a pain in the butt foor no real gains.

 

Where would you place relativity and quantum mechanics?

As mentioned above, relativiy is a subcategory of classical phyics. Quantum mechanics would be a different branch of physics than classical mechanics.

 

I should make it clear that the only reason I created this thread was to create awareness regarding what people mean when they use the term classical mechanics. It can lead to confusion. I have vague recollections of that happening in the past in physics forums on internet forums.

Edited by pmb
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Classical mechanics traditionally refers to continuous (rather than quantum) physics, which includes relativity.

I agree with you. However I don't believe that others won't. Hence this thread.

Sometimes though, I use it to mean non-relativistic mechanics.

I'm not sure what you mean. That could be taken to mean

 

non-relativistic mechanics (a) = Newtonian

 

non-relativistic mechanics (b) = Quantum Mechanics

 

I think this is where some disagreements will tend to rise.

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I'd like to make a request of you good folks to state the definition of classical mechanics as you choose to use the term, especially when knowing the precise definition ot it is critical.

 

Classical to me means that all the underlying spaces, say the configuration space Q or the phase space QP or some other useful space associated with the system are all classical spaces. That is the algebras of functions on these spaces are commutative algebras. In the case of the phase space, one thinks of the classical observables as functions in position and momenta. Functions on commutative spaces commute, by definition really.

 

One can, and should relax this slightly and consider spaces that are supermanifolds if you want to include quasi-classical fermions. So we can allow for "spaces" whose algebra of functions is not strictly commutative, but supercommutative. That is commutes up to a minus sign. (In fact you may want graded supermanifolds here, which we can talk about another time)

 

Quantum requires the use of more general noncommutative "spaces". Here the functions of position and momenta no longer commute. It is this noncommutativity that signals something "quantum". The underlying algebra of observables is a noncommutative algebra for a quantum system.

 

So, as special and general relativity deals with classical spaces they are classical theories.

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I think someone here on SFN once defined classical physics as anything that doesn't use [math] \hbar [/math]. Can't seem to remember who said that, but I like it.

 

That is a reliable way to spot if the equations you are dealing with a quantum in nature or not. But this is selling it short.

 

Consider the CCR

 

[math][x,p] = i \hbar[/math]

 

You should then think of Planck's constant, up to i and two pi as measuring the noncommutativity on the quantum phase space.

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That is a reliable way to spot if the equations you are dealing with a quantum in nature or not. But this is selling it short.

 

Consider the CCR

 

[math][x,p] = i \hbar[/math]

 

You should then think of Planck's constant, up to i and two pi as measuring the noncommutativity on the quantum phase space.

 

So you are saying that there are other non-classical frameworks in physics where planck's constant (or reduced plancks constant) is not seen in the important commutators?

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Classical to me means that all the underlying spaces, say the configuration space Q or the phase space QP or some other useful space associated with the system are all classical spaces. That is the algebras of functions on these spaces are commutative algebras. In the case of the phase space, one thinks of the classical observables as functions in position and momenta. Functions on commutative spaces commute, by definition really.

 

One can, and should relax this slightly and consider spaces that are supermanifolds if you want to include quasi-classical fermions. So we can allow for "spaces" whose algebra of functions is not strictly commutative, but supercommutative. That is commutes up to a minus sign. (In fact you may want graded supermanifolds here, which we can talk about another time)

 

Quantum requires the use of more general noncommutative "spaces". Here the functions of position and momenta no longer commute. It is this noncommutativity that signals something "quantum". The underlying algebra of observables is a noncommutative algebra for a quantum system.

 

So, as special and general relativity deals with classical spaces they are classical theories.

I wish I knew what you were talking about. :D You're describing this in some way that I don' recognize. It seems that what you wrote could also be interpreted as quantum mechanics. Can you restate it so that it makes moe sense to me? I.e. please dumb it down for me.:P

 

It seems to me to be a theoretical statement of the physics of something but I can't put my finger on it. I've learned a lot in the last 27 years and there's no way I'll ever remember all of it no matter how much refreshing I do.

Edited by pmb
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So you are saying that there are other non-classical frameworks in physics where planck's constant (or reduced plancks constant) is not seen in the important commutators?

 

In the context of quantising spaces themselves, say via a deformation or quantising the coordinate ring (a la Manin) you usually want a parameter that plays the role of Planck's constant, though the units may be different.

 

I am thinking that in the path integral formulation it is probably not immediate that Planck's constant measures the noncommutativity of something. Here it is the parameter that controls the contribution of the paths to the amplitude. The configuration space of all paths is a classical space, so noncommutativity here is harder to see.

 

I wish I knew what you were talking about. :D You're describing this in some way that I don' recognize. It seems that what you wrote could also be interpreted as quantum mechanics. Can you restate it so that it makes moe sense to me? I.e. please dumb it down for me.:P

 

In classical physics any two observables (fundamentally functions of position and momentum) commute. That is

 

[math]A(x,p)B(x,p) = B(x,p) A(x,p)[/math].

 

(Just think about multiplication of real numbers)

 

In quantum theory any two observables will in general not commute. To distinguish the quantum position and momentum I will put a hat on them. In particular we have

 

[math]\hat{x}\hat{p} - \hat{p}\hat{x} = i \hbar \hat{1}[/math],

 

which is really the root of all quantum mechanics. Non-relativistic quantum mechanics is then all about the representation theory of the above.

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In classical physics any two observables (fundamentally functions of position and momentum) commute. That is

 

[math]A(x,p)B(x,p) = B(x,p) A(x,p)[/math].

 

(Just think about multiplication of real numbers)

 

In quantum theory any two observables will in general not commute. To distinguish the quantum position and momentum I will put a hat on them. In particular we have

 

[math]\hat{x}\hat{p} - \hat{p}\hat{x} = i \hbar \hat{1}[/math],

 

which is really the root of all quantum mechanics. Non-relativistic quantum mechanics is then all about the representation theory of the above.

I'm sorry but I don't understand, why the need to make it so complex in your other post? It's just so much easier to say that Classical Mechanics is physcs which is outside of quantum mchanics? Everyone woiuld know what you meant. This thread was supposed to be about what

you think

state the definition of classical mechanics as you choose to use the term especially when knowing the precise definition of it is critical.

Did you interpret "precise" to mean that I was asking about something more complicted than

Classical physics is about Newtonian Physics + Relativity but not about quantum mechanics.

Perhaps I didn't state the question so as I'd get the type of response I was looking for. I was looking for a non-mathematical definition of "classical mechanics".

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It's just so much easier to say that Classical Mechanics is physcs which is outside of quantum mchanics?

 

Okay, that we can agree on, but...

 

Did you interpret "precise" to mean that I was asking about something more complicted than

 

I interpreted the question as looking for a fairly robust definition of classical and quantum. To my mind, the best distinction is the commutative verses the noncommutative world.

 

Perhaps I didn't state the question so as I'd get the type of response I was looking for. I was looking for a non-mathematical definition of "classical mechanics".

 

This is going to be difficult as classical mechanics is really a mathematical construct in which to model phenomena. Very hand waving, classical mechanics works well for anything that is not too small, i.e. quantum effects are negligible.

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This is going to be difficult as classical mechanics is really a mathematical construct in which to model phenomena.

I don't know what "is really" means. Is it anything like Israeli? :P

 

Classical Mechanics is a system which is based on Newton's laws. If the Galilean transformation is employed to transformed between inertial frames then we have what I refer to as Classical Newtonian Mechanics (for lack of another term) and when we use Lorentz transformations we have SR. When we add to this like equivalence principle etc the we have GR.

 

That's my definition and how precise it is depends on its applications, not the definition.

 

And that's they way PMB seems it. :)

 

ajb - What does this hand waving thing you mentioned have to do with anything? Take a look at Goldstein et al's text and notice how everything they've derived is rooted on Newton's three laws which have a very small explicit mathematical content it them. Lagrangian and Hamiltonian mechanics can be deduced from those laws as can the rest of classical mechanics.

 

Awww shucks! I did it again! I got into a discussion about a definition without being aware of it that much. I sure can be slow in the noggin at times. :(

Edited by pmb
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It's just so much easier to say that Classical Mechanics is physcs which is outside of quantum mchanics?

 

 

Okay, that we can agree on, but...

 

 

But what about mechanics that is neither quantum nor the other?

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Classical Mechanics is a system which is based on Newton's laws. If the Galilean transformation is employed to transformed between inertial frames then we have what I refer to as Classical Newtonian Mechanics (for lack of another term) and when we use Lorentz transformations we have SR. When we add to this like equivalence principle etc the we have GR.

 

 

Okay, so what are the corresponding transformations for non-relativistic quantum mechanics?

 

Well for the free Schrödinger equation we have the Schrödinger group. This group is the Galilean group with a central extension.

 

The Lorentz transformations play a very important role in quantum field theory on Minkowski space-time, so I cannot see that we can necessarily separate classical and quantum here.

 

ajb - What does this hand waving thing you mentioned have to do with anything? Take a look at Goldstein et al's text and notice how everything they've derived is rooted on Newton's three laws which have a very small explicit mathematical content it them. Lagrangian and Hamiltonian mechanics can be deduced from those laws as can the rest of classical mechanics.

 

 

It is more the other way round and one can certainly write down mechanical systems that do not obey Newton's laws.

 

In essence, by picking just about the simplest Hamiltonian or Lagrangian for a point particle (these will be non-degenerate and so equivalent) you can "re-derive" Newtonian mechanics. However, the formalism works much more generally than that.

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I think someone here on SFN once defined classical physics as anything that doesn't use [math] \hbar [/math]. Can't seem to remember who said that, but I like it.

 

Yes. Precisely the classical limit of a quantum theory is the limit when [math]\hbar \rightarrow 0[/math].

Edited by juanrga
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Yes. Precisely the classical limit of a quantum theory is the limit when [math]\hbar \rightarrow 0[/math].

 

But one should take care to note that not all quantum theories have unique classical limits. It is possible for a quantum theory to arise from the quantisation of two or more distinct classical actions.

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There were some points which I shouldn't have made being that tired :o and then , on top of that, commenting on things I have zero interest in - Mix them together and we have bad juju :P

 

At this point I lost interest (I actually had nothing more to say after my first opening post. Since there's too much other work to do this will be my point of exit. Thanks.

 

Note: I will say one last thing before I sign off this thread. When I started this thread it to get an idea of how many people use/interpret the term Classical Mechanics to include relativistic mechanics as Goldstein uses/defines it.

Edited by pmb
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Note: I will say one last thing before I sign off this thread. When I started this thread it to get an idea of how many people use/interpret the term Classical Mechanics to include relativistic mechanics as Goldstein uses/defines it.

 

I would say that just about every theoretical and mathematical physicists would.

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Isn't there a way to delete a post I make? E.g. if I post in a thread and a hour later I decide that it was unwise that I posted it at all then I'd like a way to be able to delete the post. Other forums have that function and seem to use the same software so it seems like it should be possible for the software to be adjusted to make it so.

 

 

 

I would say that just about every theoretical and mathematical physicists would.

I'm confused. Are you saying that most would go with Goldstein et al? I need to ask because earlier this morning when I first logged on I read it the other way. :doh:

Edited by pmb
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snapback.pngajb, on 19 May 2012 - 10:14 AM, said:

 

I would say that just about every theoretical and mathematical physicists would.

 

I'm confused. Are you saying that most would go with Goldstein et al? I need to ask because earlier this morning when I first logged on I read it the other way.

 

seconded.

 

 

pmb

But surely in your opening post you showed that Goldstein played the good guy by acknowledging that there are different interpretations of the phrase 'classical mechanics' and defining precisely what definition he was going to employ?

 

I repeat my comment one more time that there are yet more sorts of mechanics than quantum v classical.

 

I even offered the example of a type which is sensitive to scale (colouring of cellular automata) which is neither quantised nor does it follow normal rules.

 

go well

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