New research in classical physics

[Resha Caner]

Would anyone know if there is "new" work going on with respect to classical physics?

 

To further clarify that question, let me explain that I'm sure there is work being done to (as Kuhn would say it) further articulate the paradigm. You know of what I speak. Someone writes a paper on the general stress in flat plates. Then someone writes a paper on circular plates with fixed edges. Then someone writes a paper on circular plates with fixed edges supporting a tabby cat. Then someone writes a paper on circular plates with fixed edges supporting tabby cats with a cold ... that's not what I'm interested in.

 

I'd be more interested in work related to classical-quantum correspondence. I know there are several different directions one can go from there. The most obvious would be people seeking to unify the two. Next would be those who are attempting to show that one or the other is completely unfounded (i.e. that quantum physics is all wrong and classical physics can be used to explain everything).

 

But I'm looking to split the hair even further. Amongst those looking into such matters, have any suggested that a classical-quantum correspondence most likely needs to be achieved by modifying classical physics?

 

Let me give an example. Newton's second law (F=ma) applies to rigid bodies. If one considers deformable bodies, a stiffness term is added (F=ma+kx). If one considers how mechanical motion is dissipated as heat, damping term is added (F=ma+cv+kx). One could theoretically go on forever adding terms to capture different phenomena.

 

So, is anyone trying to extend or reformulate the laws of classical physics? If so, I'd be interested in a name so I could educate myself (or at least attempt it).

 

I'm sorry that this got so long, but I was hoping to make my question as clear as possible.

[ajb]

Would anyone know if there is "new" work going on with respect to classical physics?

 

People do work on classical mechanics in many different contexts.

 

What I am more familiar with is the geometric ideas behind classical mechanics, say symplectic & Poisson manifolds and then all the interesting things related to these.

 

 

I'd be more interested in work related to classical-quantum correspondence.

 

Quantisation of classical systems is a big industry with plenty of heavy mathematics. Roughly, you want a procedure from Poisson manifolds to Hilbert spaces. (+ some technical stuff)

 

Also, de-quantisation is a difficult topic. That is Hilbert spaces to Poisson manifolds.

 

The most obvious would be people seeking to unify the two. Next would be those who are attempting to show that one or the other is completely unfounded (i.e. that quantum physics is all wrong and classical physics can be used to explain everything).

 

Is anyone trying to unify the two? I am not aware of any work in this direction.

 

But I'm looking to split the hair even further. Amongst those looking into such matters, have any suggested that a classical-quantum correspondence most likely needs to be achieved by modifying classical physics?

 

The usual way of working to to start with a classical system and then proceed to a quantum one.

 

This is not really how one would like it to be. The universe is inherently quantum and one would like to start with a fully quantum system from the outset.

 

It depends what you mean by modification to classical physics. For example, trying to quantise general relativity suggests that there maybe extra terms in the action not required by the classical theory.

 

Another answer to your question could be that the quantum theory of fermions requires a semi-classical theory in which we have variables that anticommute.

 

This is not seen in classical mechanics.

 

One can also think about mechanics over noncommutative spaces.

 

 

 

So, is anyone trying to extend or reformulate the laws of classical physics? If so, I'd be interested in a name so I could educate myself (or at least attempt it).

 

Again, this depends on what you mean. There is great interest in geometric formulations of classical and semi-classical mechanics.

Edited November 30, 2009 by ajb

[Resha Caner]

ajb, thanks for the reply. I hope you won't regret it. I did some digging around, and found that many people are working on the edge between classical and quantum mechanics. One example would be Dr. Friedman from Amherst, but there are others.

 

My search here involves a bit of thrashing about for several reasons. First because my training is mainly in classical physics, so I only have the vaguest grasp on quantum physics. I know that if I'm to be serious about this, I need to take the plunge into quantum physics sometime. Second, I do have some ideas that I'd like to formulate, but I suspect others have thought of it before me. So, I'm digging to see if that is the case. I suppose I should have become a physics professor (rather than an engineer) so it would be my job to do this sort of thing.

 

Anyway, my point is that symplectic manifolds might be the answer I'm looking for, but since I don't know what they are, that's hard to say. And it's a daunting task to start with some elementary text and work my way up to understanding it only to find it's not what I'm looking for (I've done that several times now). I know I'm leaning on the kindness of strangers to tell me: oh yeah, that's been done by so-and-so in this-and-such field of research. Instead, you have every right to say: why is this my problem?

 

I'll try to further clarify my thoughts, and we'll see if someone can recognize that it's been done, that it's a stupid idea, or that it's just not important.

 

In Newtonian physics, "force" is the central concept. Without force, a body maintains its state of rest or uniform motion. Newtonian physics, then, becomes the study of how force changes motion.

 

Lagrange, Hamilton, etc. reformulated physics using "energy" as the central concept (or the variational principle of least action). As it turns out, it's pretty easy to show that these two formulations are equivalent.

 

Then along comes quantum physics (and I won't attempt to state it's central concepts for obvious reasons, though I know it's based on the idea that energy only occurs in discrete quanta). I also understand that quantum physics arose because of the attempt to explain the behavior seen in the micro (i.e. subatomic) world rather than the macro world. That is why it interests me.

 

I mentioned Newton's second law earlier, and how people have extended it to include deformation and dissipation. But the concept of "stiffness" (force proportional to displacement) and "damping" (force proportional to velocity) are really only mathematical conveniences. Once one begins to ask what causes stiffness, one descends into the world of materials science and then chemistry and so forth, and finds there is no good link between those sciences and the concept of "stiffness". It gets even worse if the stiffness is not linear.

 

That is when I start to ask myself: even though Newton's idea of force is convenient for a select number of linear problems, is there a better way to formulate the problem for those cases where it does not fit well?

[ajb]

Symplectic and their degenerate cousins Poisson manifold form the basis of the mathematical structure of classical mechanics. In fact this is in both the Hamiltonian and Lagrangian formulations.

 

The notion of a force gets greatly generalised in analytical mechanics. I do not think there is any real fundamental issue with things being non-linear. Of course it makes it more complicated.

[Klaynos]

I'd suggest doing some reading on Hamilton–Jacobi mechanics... And how this approaches quantum mechanics...

[Resha Caner]

I'd suggest doing some reading on Hamilton–Jacobi mechanics... And how this approaches quantum mechanics...

 

Alright. I shall do that. In the meantime, maybe I'll continue to elaborate a bit more to see if it strikes a chord with anyone.

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I do not think there is any real fundamental issue with things being non-linear. Of course it makes it more complicated.

 

Yes and no. Let me continue, because I consider nonlinear models a bandaid for systems that don't fit linear models (despite the elaborate math). Maybe that is just a philosophical issue of whether the system is "really" nonlinear, but hear me out.

 

Because nonlinear models can be solved numerically, many people have stopped looking for the underlying physics (at least within my circle). It's too easy to plug equations into a computer and let it chug away. But the chosen form for the nonlinear model is not a settled matter. I know of one paper that makes a case against a priori selection of nonlinear models and another that demonstrates how the selection affects the results. Further, there was a "definitive" paper written on nonlinear models for elastically mounting machinery. The paper actually suggested 2 options. Both options allowed for bifurcations. A subsequent paper chose yet a third option that did not allow for bifurcations. These papers were rather old by the time I came across them, so I've had difficulty contacting the authors to ask if the elimination of bifurcations was intentional. The data to which it is compared seems to imply this, but it is never explicitly stated. The issue concerns me because I have had discussions with 2 of the world's leading experts in (macro) nonlinear dynamics. One asked me if I believed bifurcations were real because he had never personally witnessed one in the lab. Another, when I asked for help in using stochastic methods, told me not to bother - the effort wasn't worth the small payoff.

 

All the frustrations I've encountered have gotten me to thinking about the limits of existing models. For example, I noticed that everyone uses displacement and the first 2 derivatives (velocity & acceleration) in their models. But no one uses the third derivative (jerk) - except to post-process data. I even found a comment in DenHartog's book that it's fruitless to integrate jerk into a dynamic model (which implies he tried it). I tried as well, and have to agree with him.

 

Or think of a mass suspended on a spring. Had the idea of an imaginary coordinate not come along, people would likely have concluded it's motion was nonlinear. After all, the motion fits pretty well to a polynomial of the form x = t - t^3/3! + t^5/5! - ... (that's a joke you hopefully get).

 

So, are we concluding certain systems are nonlinear when, if we knew how to properly represent them, we'd find they actually aren't? Or, at least, that the nonlinear behavior has a much simpler form than we tend to use. I could explain further with a pendulum example, but I'll pause here to give a chance for this to soak in.

[ajb]

Because nonlinear models can be solved numerically, many people have stopped looking for the underlying physics (at least within my circle).

 

Plenty of people are interested in non-linear science. For example plenty of people are interested in exactly solvable systems, solitons and alike.

 

 

All the frustrations I've encountered have gotten me to thinking about the limits of existing models. For example, I noticed that everyone uses displacement and the first 2 derivatives (velocity & acceleration) in their models. But no one uses the third derivative (jerk) - except to post-process data. I even found a comment in DenHartog's book that it's fruitless to integrate jerk into a dynamic model (which implies he tried it). I tried as well, and have to agree with him.

 

I expect that you can handle this and higher derivatives using jet bundles.

 

Another way is to introduce axillary variables. You should be able to reduce it down to first derivatives only. I expect this is why no-one really deals with anything past acceleration. With the exception of engineering.

 

There is then the question of quantisation with higher derivatives.

 

 

 

So, are we concluding certain systems are nonlinear when, if we knew how to properly represent them, we'd find they actually aren't? Or, at least, that the nonlinear behavior has a much simpler form than we tend to use. I could explain further with a pendulum example, but I'll pause here to give a chance for this to soak in.

 

You mean like have a linear superposition principle?

Edited November 30, 2009 by ajb

[Resha Caner]

Plenty of people are interested in non-linear science. For example plenty of people are interested in exactly solvable systems, solitons and alike.

 

Sure, they're interested. But I haven't seen many with practical solutions. I can make up a host of nonlinear equations with exact solutions for special cases. But are they applicable to a known system? As an engineer, one of the first questions I get when a crazy new idea comes along is: is it practical?

 

But, as I said, it may just be that I'm unaware of the available solutions. For example "soliton" and "jet bundle" are new terms for me. I'll look them up.

 

You mean like have a linear superposition principle?

 

Exactly - except it would be a nonlinear superposition principle. I'm aware of some math that would allow me to do that, but I struggle every time I try to apply it to a real problem. The pendulum would be an example. The solution of pendulum response to gravity is known. But (as far as I know), no exact solution has been found for anything beyond that. If you add some damping to the problem, or a periodic torque, every paper I've seen then resorts to numerical methods.

 

The problem is that the known solution is given as a Jacobi elliptic function (the elliptic sine, sn(u,k)). The k parameter is a function of the initial potential energy of the pendulum. So, if you change the energy state of the pendulum with additional force, it would mean the k parameter becomes a function of the work done on the pendulum (or a function of time). From there I don't know of an exact, closed form solution. But, as I look at the math, it seems to me to imply that I need a change in my reference (i.e. a different formulation of time) or a different formulation of force. The convenient one would be one that compliments the nonlinear superposition I would use, but I haven't been able to make that work.

 

As I said, though, there are enough smart people in the world that I figured someone must have looked at this by now. Maybe the answer lies in some of the possibilities already suggested.

[ajb]

But are they applicable to a known system? As an engineer, one of the first questions I get when a crazy new idea comes along is: is it practical?

 

My perspective is much more mathematical

 

 

Exactly - except it would be a nonlinear superposition principle.

 

Also look up Bäcklund transforms. I know they can be used to create new solutions form old. They are an important part of soliton theory. Just don't ask me for details, I would have to consult my library.

 

As I said, though, there are enough smart people in the world that I figured someone must have looked at this by now. Maybe the answer lies in some of the possibilities already suggested.

 

There is a lot of material out there relate to non-linear science. It is not a specialisation of mine, I am aware of some works and ideas but that is about it.

 

Asking about how to generalise mechanics does intersect with my work and interests. I have done some work on generalising Poisson geometry. You can find a thread on the subject in this forum.

 

Something else I am also interested in is classical mechanics and the BV formalism on Lie algebroids. I have made some initial study in this direction. In particular I have developed the description of Lie algebroids in terms of double vector bundles in the category of graded manifolds. It gives me all the brackets I need to develop what I am interested in. Well, plenty of work left.

 

See http://xxx.soton.ac.uk/abs/0910.1243

[Resha Caner]

Asking about how to generalise mechanics does intersect with my work and interests.

 

Well, then I'll be patient and wait for you to solve my problem.

 

If I had the resources, I'd fund someone to help me do it ... Maybe someday.

[ajb]

Something else I should have pointed out to you is that Gozzi and Reuter (circa 1988) have considered how to formulate classical mechanics in terms of path-integrals. I am sure by now other people have also looked at it.

 

I believe their work has something to do with the BV-antifield formalism and how this related to the underlying symplectic geometry of classical mechanics. It is definitely on the list of things to get familiar with.