Talk:Hamiltonian mechanics
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)
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?
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 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 :-
![{\displaystyle {\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]}](/media/api/rest_v1/media/math/render/svg/a0c23d2027a7c13dbf38f7d01c7fc9c5f99f7959)
and on substituting Hamilton's equations for terms 2 & 3 we get the Poisson Bracket of f & H plus the partial time derivative.
So 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 anyway!
I'm being over-polite - whoever wrote it (appears to me to have) got it round their neck.
Bob
Linuxlad 11:30, 10 Nov 2004 (UTC)