Talk:Hamiltonian mechanics

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Contents

[edit] Intro

Hmm, wouldn't an introduction to Hamiltonian mechanics WITHOUT starting from Lagrangians and starting from symplectic spaces and Poisson brackets be more natural? Phys 19:09, 5 Sep 2003 (UTC)

Er, NOT to most physicists chemists and engineers... :-) Linuxlad 19:01, 17 Mar 2005 (UTC)

[edit] Page move

The move of page title isn't a good idea. We generally prefer general titles (Hamiltonian mechanics), to more special ones, such as particular equations.

Charles Matthews 10:36, 16 Jun 2004 (UTC)

For what it's worth, I agree with Charles. Also, can someone please explain the current mess of Talk pages involved? Especially Talk:ȧ£æžåŠ›å... (Mozilla won't let me type the whole of it).

Taral 17:42, 17 Jun 2004 (UTC)

What I want to know is why the Hamiltonian view of mechanics is more useful than the classical version? What can engineers and scientists do with it that they cannot do w/o it?

[edit] Norm

Liouville's equation here shows total d/dt equal to the Poisson bracket. A physicist would expect to write partial d/dt here, because the essence of Liouville is that total d/dt, meaning the convective derivative taken with the particle, is zero. See main Liouville's theorem (Hamiltonian)

Linuxlad 12:28, 9 Nov 2004 (UTC)

Looking at this issue again, Goldstein's Classical Mechanics

(Later Note - this is Goldstein 1964 edition ie 1950 2nd reprint)

at the ready, there appear to be two differences from what I'd expect:-

In the absence of any further constraints on f, I'd expect (cf Goldstein eqn 8-58) that:-

a) the convective/total time derivative of f equalled the Poisson bracket of f & H PLUS the partial time derivative of f.

b) In the _particular_ case of phase space density (or probability) it is possible to show that the convective derivative is zero, so that the partial derivative equals minus the Poisson bracket (Goldstein eqn 8-84) - but note that this result does NOT follow trivially from the result for general f as implied.

(So I reckon that's 2/3 violations of my naive physicist's expectations) - I hereby give notice that I may edit accordingly :-)

Linuxlad 10:18, 10 Nov 2004 (UTC)

This page certainly needs some work. For example it doesn't give (and neither does the page linked to) the classical expression of the Poisson bracket. As far as I can see, though, the definitions are the standard ones, such as are given in Abraham and Marsden, Foundations of Mechanics, though.

Charles Matthews 11:08, 10 Nov 2004 (UTC)


Just to be clear, I'm looking at the first part of the section entitled Mathematical formalism. and specifically the two equations for f and for ρ

So

Equation 1 For general f :-

\frac{df }{dt}=\frac{\partial f }{\partial t}+\sum_{i=1}^{N}\left[ \frac{\partial f }{\partial q_{i}}\dot{q}_{i}+\frac{\partial f }{\partial p_{i}}\dot{p}_{i}\right]

and on substituting Hamilton's equations for terms 2 & 3 we get the Poisson Bracket of f & H plus the partial time derivative.

So, (from standard maths methods viewpoint), first equation has partial df/dt missing.


2nd equation (for ρ) - Liouville's theorem is TOTAL d by dt of phase space density is zero, which does NOT follow directly from above, and is not what's given there anyway (you need to change total time derivative to partial)!

I have now edited acccordingly

Bob

Linuxlad 11:30, 10 Nov 2004 (UTC)

[edit] Hamiltonian systems

There is a redirection from "hamiltonian system" to this page, which is certainly better than nothing, but not really satisfying in my opinion of view.

There should be Hamiltonian systems, integral of motion, integrable systems, Lax pairs, ... Amateurs, please try to create some stubs... MFH 17:48, 17 Mar 2005 (UTC)

I utterly agree. A Hamiltonian system is never defined anywhere. I vaguely remember what it is from my lectures ten years ago but that's not enough. I try a stub please correct me! --131.220.68.177 16:34, 19 August 2005 (UTC)

[edit] Classical Relativistic Electromagnetic Hamiltonian

This article seems to have sprouted a section:

Classically (non-quantum mechanical), for a particle of charge q and mass m in an electromagnetic field with vector potential A and scalar potential φ, the relativistic Hamiltonian is
H = \sqrt{(\mathbf{p}-q\mathbf{A}/c)^2 c^2 + m^2 c^4} + q\phi
where the term qφ is potential energy, the radical term is total energy of the particle, in terms of the energy-momentum relation, with momentum replaced by what may be construed as a "gauge covariant momentum": it is called kinetic momentum.

However, I don't beleive this has anything to do with hamiltonian mechanics. Well, its an example of a hamiltonian, but its one of a thousand examples, and doesn't seem to be a particularly good one. If the goal is to add examples, something simpler should be given: e.g. ball on an inclined plane or something like that. Perhaps this belongs in its own article? linas 00:16, 26 October 2005 (UTC)

Don't fully agree with the above. Formulating the classical Hamiltonian for a particle in an e/m field, is a standard way (eg 3rd year undergraduate level, I recollect) of working to the important quantum mechanical case. Linuxlad 11:14, 11 November 2005 (UTC)
We have distinct articles for things like the simple harmonic oscillator and the hydrogen atom. I strongly urge you to start an independent article on the --classical relativistic electromagnetic Hamiltonian or maybe relativistic charged particle. Yes, I am well aware that this Hamiltonian is treated in many books, including the awe-inspiring development in Landau and Lifschitz. However, Hamiltonian mechanics has many, many facets to it that stretch far beyond what the third-year undergraduate is aware of. A distinct article for this will allow it to be expanded as it deserves (after all, L&L found 100+ pages of things to say about it). linas 22:36, 18 November 2005 (UTC)

I don't think you've made your case here! The example neatly extends the idea from the pure mechanical variables into e/m using probably the most important example of a single-particle system. But perhaps you're right - this is really a page for people whose searching for deeper mathematical insights should not be sullied with examples of the real world :-) Linuxlad

I frankly don't understand why we are arguing about this. There are thousands of Hamiltonians, and while relativistic charged particles are interesting and important for many reasons (and deserve their own article (hint hint)), I can't imagine why they are "important" for hamiltonian mechanics. Surely a ball running down an inclined plane is a "more important" example (as it illustrates contact geometry, sub-Riemannian geometry and lagrange multipliers all at the same time? Maybe a pendulum, as that can illustrate the transition to chaos, the KAM torus and the phase-locked loop at the same time? The lorenz attractor? An example from fluid dynamics? Why charged particles?
As to the quality of this article; at this current time, it is rather incomplete at this time. However, that's a different issue. linas 01:13, 19 November 2005 (UTC)
(Have you ever read the Landau and Lifshitz treatment of the classical relativistic charged particle?) linas 01:19, 19 November 2005 (UTC)
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