- #1

- 214

- 1

- Thread starter eljose79
- Start date

- #1

- 214

- 1

- #2

pmb

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.Originally posted by eljose79

Pete

- #3

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

- #4

pmb

Why do you think a non-holonomic system does not have a Hamiltonian?Originally posted by selfAdjoint

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

Pete

- #5

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

"...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

(Emphasis in original).

- #6

pmb

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 sayOriginally posted by selfAdjoint

"...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 forholonomic systems only"

(Emphasis in original).

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 isIn this example, the constraint of "rolling" is actually holonomic ...

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

- #7

jeff

Science Advisor

- 658

- 1

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 eljose79

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.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...

Last edited:

- #8

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

- #9

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

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.

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

Last edited by a moderator:

- #10

pmb

Thank you very much.Originally posted by selfAdjoint

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.

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

Last edited by a moderator:

- #11

pmb

Do you know if the part you mention from Greenwood's text is also found in Goldstein and/or Lanczsos?Originally posted by selfAdjoint

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

Thanks

Pete

Last edited by a moderator:

- #12

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

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).

- #13

pmb

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.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).

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

- #14

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

- #15

pmb

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?Originally posted by selfAdjoint

Pete

- #16

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

- #17

pmb

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.Originally posted by selfAdjoint

How are you doing through all of this?

Pete

- #18

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

- #19

pmb

I'm sort of a poster child for howOriginally posted by selfAdjoint

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

- #20

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,786

- 7

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

- Last Post

- Replies
- 2

- Views
- 4K

- Last Post

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 2

- Views
- 5K

- Last Post

- Replies
- 2

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 3K

- Last Post

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 12

- Views
- 1K

- Last Post

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 6

- Views
- 2K

- Last Post

- Replies
- 25

- Views
- 5K