# Quantization without Lagrangians.

#### eljose79

In fact let us suppose we only have the classical equations of movement x´=f(x1,x2,x3...xn) but we do not have or not know a lagrangian ..how could we quantizy them?..in fact how is a quantization made if we do not have a lagrangian (or hamiltonian)?..

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#### pmb

Originally posted by eljose79
In fact let us suppose we only have the classical equations of movement x´=f(x1,x2,x3...xn) but we do not have or not know a lagrangian ..how could we quantizy them?..in fact how is a quantization made if we do not have a lagrangian (or hamiltonian)?..
I suppose that you can observer the system and then ask what Lagrangian produces a Hamiltonian which will describe the behaviour of the system. I don't know if its possible in practice though.

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Most systems have a lagrangean. Some systems do not have a Hamiltonian, they are called nonholonomic (or anholonomic for purists). The great example is a wheel rolling on a flat surface.

#### pmb

Most systems have a lagrangean. Some systems do not have a Hamiltonian, they are called nonholonomic (or anholonomic for purists). The great example is a wheel rolling on a flat surface.
Why do you think a non-holonomic system does not have a Hamiltonian?

See - http://www.cds.caltech.edu/~koon/papers/nonholonomic_1.pdf

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This work is new to me. What I stated was the standard info from mechanics. For example, Greenwood's "Classical Dynamics", page 161:

"...hence we conclude once again that operations of variation and integration can be interchanged for the case of holonomic systems, but this is not possible for nonholonomic systems.
We also conclude that Hamilton's principle,...,is valid for holonomic systems only"

(Emphasis in original).

#### pmb

This work is new to me. What I stated was the standard info from mechanics. For example, Greenwood's "Classical Dynamics", page 161:

"...hence we conclude once again that operations of variation and integration can be interchanged for the case of holonomic systems, but this is not possible for nonholonomic systems.
We also conclude that Hamilton's principle,...,is valid for holonomic systems only"

(Emphasis in original).
That is strange. I can't understand why they'd say that. A disk rolling on a plane clearly has a Hamiltonian. Nonholonomic systems are usually solved by the method of Lagrange undetermined multipliers. This is coevered in Goldstein's et al's text "Classical Mechanics - 3rd Ed." It's in the section labeled "Extension of Hamilton's Principle to Nonholonomic systems." As an example of this method they use a hoop rolling down an inclined plane as an example. Strangely enough they say
In this example, the constraint of "rolling" is actually holonomic ...
The solution to this problem is the same as the loop rolling in a plane - just set the incline angle to zero. The Lagrangian is then simply the kinetic energy which is

L = (1/2)M(dx/dt)^2 + (1/2)Mr^2 [d(theta)/dt]^2

Then us the usual Legendre transformation to get the Hamiltonian.

But in any case - Do you know of a Lagrangian for which no Hamiltonian can be defined? I can't imagine why since the Hamiltonian is defined in terms of the Lagrangian by calculating the energy function h defined as

h = dq/dt p - L

h is a function of qdot, q and t at this point. Calculate the canonical momentum p as p = &L/&qdot. Then solve for qdot in terms of p, q and t and substitute into the energy function. The result is the Hamiltonian. If this procedure can be done then there is a Hamiltonian.

The Hamiltonian for the loop roplling down the plane is a homework problem in Goldstein - Chapter 8 problem 17.

The complete referance to the paper I mentioned above is

"The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems," Koon, W. S. and J. E. Marsden, Rep. Math. Phys., 40, 21-62

Pete

#### jeff

Originally posted by eljose79
In fact let us suppose we only have the classical equations of movement x´=f(x1,x2,x3...xn) but we do not have or not know a lagrangian ..how could we quantizy them?..in fact how is a quantization made if we do not have a lagrangian (or hamiltonian)?..
Looking at the general case of QFT, there's no proof that every conceivable QFT must be formulated in terms of the canonical quantization of some lagrangian: It's possible that there may be QFTs that lead to physically satisfactory S-matrices that can't be derived by the canonical quantization of some lagrangian. But if you've got the equations of motion, you should be able to construct a lagrangian whose variation it's derived from.

Originally posted by pmb
Do you know of a Lagrangian for which no Hamiltonian can be defined? I can't imagine why since the Hamiltonian is defined in terms of the Lagrangian...
If you've got a lagrangian, the hamiltonian obtained as it's legendre transform allows the definition of an S-matrix that is automatically lorentz-invariant. However, one may choose hamiltonians which cannot be obtained as the legendre transform of any lagrangian, but it's very tricky to obtain hamiltonians in this way that lead to S-matrices that are physically satisfactory.

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pmb - odd indeed. Greenwood is careful to restrict his Legendre transform to holonomic systems, and once again the italics are his. The original copyright is 1977.

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Some neww info. Apparently the formation of Hamiltonians for nonholonomic constraints results from new research done in the 1980s and 1990s. Systems on Riemannian manifolds and bundle-theoretic analysis were involved.

Here is a http://www.imaff.csic.es/mat/david/noholointeg.PDF on the new work.

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#### pmb

Some neww info. Apparently the formation of Hamiltonians for nonholonomic constraints results from new research done in the 1980s and 1990s. Systems on Riemannian manifolds and bundle-theoretic analysis were involved.

Here is a http://www.imaff.csic.es/mat/david/noholointeg.PDF on the new work.
Thank you very much.

When an interesting topic like this one comes up I tend to post it on more than one forum so that I get a wider variety of opions. But I have to say - I'm getting sick and tired of that damn sci.physics newsgroup. The moment you question the validity of something - textbook or otherwise - the flaming automatically begins. What jerks!

I always question a concept if I don't understand it. More so lately since in recent years I've done some proof reading of a few physics texts and as such I had some concrete examples of how and why errors get into texts.

I'd love to know why it is that some people just love to insult others when unprovoked. Such behavior is beyond my comphrehnsion.

Pmb

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#### pmb

Some neww info. Apparently the formation of Hamiltonians for nonholonomic constraints results from new research done in the 1980s and 1990s. Systems on Riemannian manifolds and bundle-theoretic analysis were involved.

Here is a http://www.imaff.csic.es/mat/david/noholointeg.PDF on the new work.
Do you know if the part you mention from Greenwood's text is also found in Goldstein and/or Lanczsos?

Thanks

Pete

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No, Greenwood (a Dover reissue) is the only good mechanics book I have now. I think I have some of my daughter's texts from college, but they are light and wouldn't cover it.

I do suspect the technique of merrily using the Legendre transform to construct a Hamiltonian for a nonholomic Lagrangean. That's just formalism and you couldn't depend on what you got to be real.

If you google on nonholonomic, and still more so on anholonomic - you get a lot of subscientific stuff from cranks and robotics folks. It will illustrate that the problem I mentioned was a truly felt one a decade or so ago.

About sci.physics, I don't know if you mean that specifically or sci.physics.research. I have had good results with the latter, but then I tend to post questions. I find that sci.physics, sci.physics.relativity and sci.physics.particle are overrun with cranks (even by the standards of the places I usually hang out).

#### pmb

About sci.physics, I don't know if you mean that specifically or sci.physics.research. I have had good results with the latter, but then I tend to post questions. I find that sci.physics, sci.physics.relativity and sci.physics.particle are overrun with cranks (even by the standards of the places I usually hang out).
Thou speaketh the truth. Some seem so bad as to deserve the name "evil." There's one low life that loves to get his jollies bvy trying to flame me using the fact that I'm disabled. He post stuff like "get a job .. stop being a burden on massachusettes.." etc.

I mean he's not even a flamer that knows what he's talking about since SSDI is payed by the feds and it's nobodies money except mine to begin with.

But he probably knows that - but he can't find anything legitimate to use to flame.

I guess you're wise in staying away from the unmoderated newsgroups. To many wackos there.

Pete

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Pete, my wife is on Disability (Congestive Heart Syndrome) and it's been a godsend to us. Anyone who begrudges it is a louse.

#### pmb

Pete, my wife is on Disability (Congestive Heart Syndrome) and it's been a godsend to us. Anyone who begrudges it is a louse.
Thanks. Nice to know that there are decent people out there. In my case it was Acute Myloid Leukemia (AML). How's your wife doing? Is this a temporary thing - I hope?

Pete

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She broke her hip in March, had three operations, picked up a resistent infection in the process, went into heart failure and has been on a ventilator for two going on three weeks now. She is now doing much better, but the infection is still bad, and they are still working to get her to breathe on her own again. She's looking at weeks or months of rehab. So far medicare and her medigap policy are handling the cost.

#### pmb

She broke her hip in March, had three operations, picked up a resistent infection in the process, went into heart failure and has been on a ventilator for two going on three weeks now. She is now doing much better, but the infection is still bad, and they are still working to get her to breathe on her own again. She's looking at weeks or months of rehab. So far medicare and her medigap policy are handling the cost.
Wow! That's pretty rough. I'm very sorry to hear that SA. Just goes to show that no matter how bad things are they can be worse. I guess I should count my blessings for doing so well in chemo.

How are you doing through all of this?

Pete

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It has given me a new understanding of the phrase "one day at a time". I imagine you can relate. But your problems hit me pretty hard too. How has your chemo worked? Are you out of the woods yet?

#### pmb

It has given me a new understanding of the phrase "one day at a time". I imagine you can relate. But your problems hit me pretty hard too. How has your chemo worked? Are you out of the woods yet?
I'm sort of a poster child for how well someone can do. My Leukemia social worker said that I shouldn't expect to remain out of the hospital long in between chemo's since infections etc. will bring me back in. But it only happened twice - once for a touch of pneumonia and one for a nose bleed that wouldn't stop (low platelets). After it was over I asked her about it and she said I beat all the odds. She's never seen anyone do as well as I did.

But I'm not past the 5-year mark yet. It's only been 3 years. So I'm still in the danger zone. But I've chosen to not get sick again. :-)

Pete

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But I've chosen to not get sick again. :-)
Great attitude! Go for it!

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