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I have a master in math. I understand that Euler-Lagrange ODE is the result of solving the Gateaux derivative of a functional to zero. So I can see how the Euler-Lagrange ODE is useful for solving, e.g., optimal control problems. I seem to not know what the purpose of the Euler-Lagrange ODE is outside of minimizing a functional, and how this is useful for modelling, though I tried several times. I knew before reading this wikipedia article that the Hamiltonian principle is an equivalent to the Lagrangian principle. And I do understand that the Lagrangian principle relates to the Euler-Lagrange ODE. Now after reading this wikipedia article I do know exactly as much as I did before, hence, these questions remain unanswered by the article:
1) what is the principle exactly? e.g., an equation? If so, what is the recipe for composing it?
2) what is the problem that the principle solves?
3) how does the principle solve the problem?
4) how does Lagrangian principle solve the problem? (or refer to where lagrangian's principle is applied to answering 1-4; alas, it is no shame to reproduce info from another page to have a self-contained presentation...)
5) give an example without short-cuts. E.g., you pose the hamiltonian but do not explain which procedures you followed to pose it.
Surely, it is no-one's job to make the article useful to anyone. And I see it is frustrating to see that it appears useless even to optimally prepared readers at the present stage. But I want to encourage everyone who is able to make it accessible. — Preceding unsigned comment added by 2A02:C7F:EACB:CE00:6892:E17F:5044:49D6 ( talk) 17:45, 17 November 2021 (UTC)
It would be great to give the nomenclature, complete with SI base units - that would help general readers understand the article. It's probably a grat textbook summary for the specialist but surely someone can at least give an introduction for the reasonably-educated generalist ? — Preceding unsigned comment added by 51.219.114.246 ( talk) 12:00, 10 July 2023 (UTC)
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)
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?
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 :-
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)
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)
This article seems to have sprouted a section:
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)
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
The result was merge sbandrews ( t) 18:51, 23 June 2007 (UTC)
This article is supposed to be the category leader for Hamiltonian mechanics. But it offers no orientation for a general reader coming in who has never heard of either Hamiltonians or Lagrangians.
Such a reader might have been directed here by Equipartition theorem for instance (currently the object of scientific peer review at Wikipedia:Scientific_peer_review/Equipartition_theorem)
My belief is that this would be best fixed by merging in most of the content from Hamilton's equations at the top of the article, to give an overview, and an idea of where we're trying to get to, and why; before we get into the algebra fest.
The two articles appear to cover the same subject, namely what are Hamilton's equations, and where do they come from, and what new view of the world do they give one.
So it seems to me a fine idea to merge them, and gather together the best content from each. Jheald 14:09, 3 April 2007 (UTC)
Isn't the form of Lagrange's equations given there plain wrong? Why is the generalized force entering the equation at all? 131.215.123.218 ( talk) 23:03, 10 June 2008 (UTC)
One of the subsections on this page is titled "Basic physical interpretation, mnemotechnics". Can anyone explain the relevance of mnemotechnics to this section? AstroMark ( talk) 13:39, 15 August 2008 (UTC)
Would it be useful to provide an example or two, to demonstrate how Hamiltonian mechanics is used, and what insight they provide? Perhaps the examples from Lagrangian mechanics could be brought over and redone, highlighting the difference between the two. I would do it, but I don't really know the answers to my own questions. User:!jim talk contribs 22:45, 13 December 2008 (UTC)
I again removed an erroneous statement added to the section Hamiltonian mechanics#Charged particle in an electromagnetic field by DS1000 ( talk · contribs). He said "The above remains valid< ref> http://academic.reed.edu/physics/courses/Physics411/html/411/page2/files/Lecture.10.pdf< /ref> at relativistic speeds with the observation that: " (I broke the "ref" to keep this in-line). The erroneous part of this statement is that the "above remains valid" which is only true for but not for the remainder of the section which is non-relativistic. The reference given uses a non-conventional definition of the dot (time derivative) which makes it the derivative with respect to proper time rather than coordinate time. It is possible that he got the erroneous formulas from the part of the reference which contains (10.36) which is a different (from the usual relativistic Hamiltonian) which the author calls "functionally equivalent". But that is for his special purpose. Please do not re-add this material without discussing it here first. JRSpriggs ( talk) 05:12, 18 December 2008 (UTC)
First, there are no sources cited for this section, and second, shouldn't this be in the article on Hamilton's principle? —Preceding unsigned comment added by 72.10.133.106 ( talk) 01:15, 12 January 2009 (UTC)
Hello. You have mistake in the article - the right equation is that the total derivative of H by time is minus the partial derivative of L by time. 23:31, 10 February 2009 (UTC) —Preceding unsigned comment added by ירון ( talk • contribs)
Am I blind or isn't there any article about classical hamiltonian itself (a function)? Magdulewicz ( talk) 13:49, 23 June 2010 (UTC)
Does anyone know the origin of the abbreviations T and V that are "traditionally" used for kinetic and potential operators? The closest guess I can come up with is translational T, but I haven't a clue for V. Laburke ( talk) 21:07, 12 January 2011 (UTC)
This is really nitpicky. I'm fairly sure you can prove that the total derivative of the hamiltonian equals the negative of the partial derivative of the lagrangian. I don't mean to be weird about this because I look at the proof given in the article, and I'm convinced of the statement about the partial derivatives of the two equaling each other. Though, I'm pretty sure it is actually the way that I have written it here. This is a much deeper statement. I think this should be changed :). Anybody interested in finding a source for this?
-- 24.7.197.142 ( talk) 08:49, 3 February 2011 (UTC)
Somewhere in the article it is written: "One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinatizations of the same symplectic manifold."
As an engineer I have no idea what a symplectic manifold is. It makes it rather difficult when one is trying to refresh one's ideas about Hamiltonians that one should now have to go and try and understand what "Symplectic manifolds are". Please - When writing articles proceed from the simple to the complex, from the concrete to the abstract and from the particular to the general and not vice versa Phillip ( talk) 06:04, 21 April 2011 (UTC)..
I'm and economist and I was looking for an economic interpretation for Hamiltonian. I didn't found anything useful here, but I did find a classic paper on the subject, " Introduction to Hamiltonian Dynamics in Economics". Do you think this should be a new section in this article? I think that hamiltonian function shouldn't redirect tho Hamiltonian mechanics, so other areas would only look for the use of Hamiltonian function. Best, Luizabpr ( talk) 17:19, 4 July 2011 (UTC)
I have changed the subsection Notes under Simplified overview of uses, tagged by Wikipedia:No original research:
I really do not think the current version adds anything helpful to the article. All that needs to be said (if anything) is:
This has replaced that section with no heading (no need) and the tag has been removed. Furthermore
-- F = q( E + v × B) 12:25, 16 February 2012 (UTC)
The section Geometry of Hamiltonian systems looks accurate enough, with proper links, but could surely be beefed up a little bit, perhaps with a diagram or figure with all the "ingredients". YohanN7 ( talk) 14:11, 5 October 2012 (UTC)
I don't really see the point of this section. That's a lot of evolved math for nothing but Hamiltonian systems (in the sense of physics). Maybe the symplectic point of view is already enough for the interested (mathematical) physicist. Just say, a Hamiltonian system can be considered as a symplectic manifold together with a smooth function $H$ on it. The manifold, which is usually the contangent bundle of a configuration space manifold $Q$, is symplectic, because the symplectic structure gives us a notion of poisson bracket and thus we can write down Hamilton's eq. of motion as a global flow equation (give that equation + explain). If needed, one can then go over to the time dependent case and set $\omega = d \theta - d E \wedge dt$ as well as $H_{new}= H_{old} - E$, where we take the cotangent bundle of $Q \times \R$. Also this "which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian." does not make sense. Yes, there is a natural symplectic structure (the cartan derivative of the canonical 1-form), but it is not a function, let alone the Hamiltonian. 94.211.47.103 ( talk) 20:44, 10 November 2013 (UTC)
I believe this article does not has, neither explain what is, the Hamiltonian mechanics' mathematical formalism. There is no section actually presenting the main mathematical premisses of Hamiltonian mechanics.
For example:
Another missing part in the article relates with the difference between Hamiltonian mechanics and classical mechanics...
I recently made a modification on the "overview of uses", which I believe fixes some of these problems. It was reverted based on an argument of "breaking < math >'s", which I believe is far from pragmatic (besides, every < math > was correct). I'm going to wait a bit until the user who reverted it to comment here, I'm sure the leading argument is not the < math >. In case of no response, I will revert my contribution back.
The introduction of this article is way too advanced for me. I am an educated amateur not a mathematician. How about an introduction ( or a separate article) along the style of "Hamiltonian Mechanics for Dummies". I got interested in HM from reading about Newton and I am curious about the trail from Newtonian mechanics onwards. My friends agree with me that many of the maths pages on Wiki have no suitable lead in section. Whoever thought of the idea "books for Dummies" was on to a great idea. — Preceding unsigned comment added by 118.208.122.190 ( talk) 22:23, 15 March 2014 (UTC)
I think that the Heisenberg Hamiltonian is confusing. If we take the coordinates $x,y,z$ like in the Heisenberg group page, it should be proportional to . I think the one that was meant in the article is the Hamiltonian in an non-holonomic basis, but in this case have nothing to do with the corresponding momenta for coordinates , which is some what confusing. — Preceding unsigned comment added by 147.122.44.52 ( talk) 14:55, 24 July 2018 (UTC)
Section Charged particle in an electromagnetic field says that the formulas are for Lorenz gauge. I believe the Lagrangian is valid for all gauges? Can someone confirm? Ponor ( talk) 14:43, 6 May 2020 (UTC)
2020-10-27 edit: this was discussing the 2020-05-06 version of the article, where the Hamiltonian was said to be valid in Lorenz gauge, as if it would be different in other gauges for the vector potential. This has been corrected since. Ponor ( talk) 12:41, 27 October 2020 (UTC)
@ Lantonov: In the section Hamiltonian mechanics#Deriving Hamilton's equations, it says "Since this calculation was done off-shell, one can associate corresponding terms from both sides of this equation to yield:". You added a "clarification needed" template to this. Off-shell simply means that the system may be following a path which is not the one required by classical physics, i.e. not the one which extremalizes the action. It does not have to have anything to do with quantum mechanics. JRSpriggs ( talk) 17:59, 26 October 2020 (UTC)
Another, smaller, problem (more like nit-picking) is that the Lagrangian in most popular textbooks and in the related Wiki articles, such as Lagrangian mechanics, Maupertuis's principle, Principle of least action, Calculus of variations, Hamilton's principle, etc., is denoted just while in this article it is denoted . In the same way, is more popular than (e.g. Schrödinger equation). Lantonov ( talk) 07:23, 27 October 2020 (UTC)
Another incongruence:
In the overview it is written that Hamilton's equations are
referring to a relative obscure textbook of Hand and Finch.
On the other hand when deriving Hamilton's equations later, they are
In the beginning we have two Hamilton's equations, and after derivation they become three. Which is true? I think that two is the right number, at least my textbook (Landau & Lifshitz) says so. Lantonov ( talk) 08:44, 27 October 2020 (UTC)
Relativity requires the more general form in which dependence on t is explicit because you must have "background" fields or constraining objects UNLESS you go whole-hog to a generally covariant theory of everything (which does not exist at present).
In other words, anything done using the Hamiltonian formalism must (for the time being) be an approximation based on non-relativistic assumptions.
If q0 = t, then p0 = − H.
JRSpriggs (
talk)
20:38, 28 October 2020 (UTC)
So called Energy based model are applied in machine learning, essentially usage of Hamiltonian mechanics. And redirect the page Energy based models to this subsection. -- mcyp ( talk) 18:39, 5 February 2022 (UTC)
I have created a new section to detail the case of which is taught in engineering and science courses. This follows on from the section Hamiltonian mechanics § Basic physical interpretation to give more precision and detail, including proofs and citations.
The section also somewhat relates to Lagrangian mechanics § Energy, which has been getting quite a few edits lately.
Are there are any comments about this new section, or anything that anyone would like me to address? DrBrandonJohns ( talk) 02:00, 22 February 2024 (UTC)
This is the
talk page for discussing improvements to the
Hamiltonian mechanics article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
![]() | This ![]() It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||||||||
|
I have a master in math. I understand that Euler-Lagrange ODE is the result of solving the Gateaux derivative of a functional to zero. So I can see how the Euler-Lagrange ODE is useful for solving, e.g., optimal control problems. I seem to not know what the purpose of the Euler-Lagrange ODE is outside of minimizing a functional, and how this is useful for modelling, though I tried several times. I knew before reading this wikipedia article that the Hamiltonian principle is an equivalent to the Lagrangian principle. And I do understand that the Lagrangian principle relates to the Euler-Lagrange ODE. Now after reading this wikipedia article I do know exactly as much as I did before, hence, these questions remain unanswered by the article:
1) what is the principle exactly? e.g., an equation? If so, what is the recipe for composing it?
2) what is the problem that the principle solves?
3) how does the principle solve the problem?
4) how does Lagrangian principle solve the problem? (or refer to where lagrangian's principle is applied to answering 1-4; alas, it is no shame to reproduce info from another page to have a self-contained presentation...)
5) give an example without short-cuts. E.g., you pose the hamiltonian but do not explain which procedures you followed to pose it.
Surely, it is no-one's job to make the article useful to anyone. And I see it is frustrating to see that it appears useless even to optimally prepared readers at the present stage. But I want to encourage everyone who is able to make it accessible. — Preceding unsigned comment added by 2A02:C7F:EACB:CE00:6892:E17F:5044:49D6 ( talk) 17:45, 17 November 2021 (UTC)
It would be great to give the nomenclature, complete with SI base units - that would help general readers understand the article. It's probably a grat textbook summary for the specialist but surely someone can at least give an introduction for the reasonably-educated generalist ? — Preceding unsigned comment added by 51.219.114.246 ( talk) 12:00, 10 July 2023 (UTC)
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)
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?
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 :-
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)
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)
This article seems to have sprouted a section:
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)
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
The result was merge sbandrews ( t) 18:51, 23 June 2007 (UTC)
This article is supposed to be the category leader for Hamiltonian mechanics. But it offers no orientation for a general reader coming in who has never heard of either Hamiltonians or Lagrangians.
Such a reader might have been directed here by Equipartition theorem for instance (currently the object of scientific peer review at Wikipedia:Scientific_peer_review/Equipartition_theorem)
My belief is that this would be best fixed by merging in most of the content from Hamilton's equations at the top of the article, to give an overview, and an idea of where we're trying to get to, and why; before we get into the algebra fest.
The two articles appear to cover the same subject, namely what are Hamilton's equations, and where do they come from, and what new view of the world do they give one.
So it seems to me a fine idea to merge them, and gather together the best content from each. Jheald 14:09, 3 April 2007 (UTC)
Isn't the form of Lagrange's equations given there plain wrong? Why is the generalized force entering the equation at all? 131.215.123.218 ( talk) 23:03, 10 June 2008 (UTC)
One of the subsections on this page is titled "Basic physical interpretation, mnemotechnics". Can anyone explain the relevance of mnemotechnics to this section? AstroMark ( talk) 13:39, 15 August 2008 (UTC)
Would it be useful to provide an example or two, to demonstrate how Hamiltonian mechanics is used, and what insight they provide? Perhaps the examples from Lagrangian mechanics could be brought over and redone, highlighting the difference between the two. I would do it, but I don't really know the answers to my own questions. User:!jim talk contribs 22:45, 13 December 2008 (UTC)
I again removed an erroneous statement added to the section Hamiltonian mechanics#Charged particle in an electromagnetic field by DS1000 ( talk · contribs). He said "The above remains valid< ref> http://academic.reed.edu/physics/courses/Physics411/html/411/page2/files/Lecture.10.pdf< /ref> at relativistic speeds with the observation that: " (I broke the "ref" to keep this in-line). The erroneous part of this statement is that the "above remains valid" which is only true for but not for the remainder of the section which is non-relativistic. The reference given uses a non-conventional definition of the dot (time derivative) which makes it the derivative with respect to proper time rather than coordinate time. It is possible that he got the erroneous formulas from the part of the reference which contains (10.36) which is a different (from the usual relativistic Hamiltonian) which the author calls "functionally equivalent". But that is for his special purpose. Please do not re-add this material without discussing it here first. JRSpriggs ( talk) 05:12, 18 December 2008 (UTC)
First, there are no sources cited for this section, and second, shouldn't this be in the article on Hamilton's principle? —Preceding unsigned comment added by 72.10.133.106 ( talk) 01:15, 12 January 2009 (UTC)
Hello. You have mistake in the article - the right equation is that the total derivative of H by time is minus the partial derivative of L by time. 23:31, 10 February 2009 (UTC) —Preceding unsigned comment added by ירון ( talk • contribs)
Am I blind or isn't there any article about classical hamiltonian itself (a function)? Magdulewicz ( talk) 13:49, 23 June 2010 (UTC)
Does anyone know the origin of the abbreviations T and V that are "traditionally" used for kinetic and potential operators? The closest guess I can come up with is translational T, but I haven't a clue for V. Laburke ( talk) 21:07, 12 January 2011 (UTC)
This is really nitpicky. I'm fairly sure you can prove that the total derivative of the hamiltonian equals the negative of the partial derivative of the lagrangian. I don't mean to be weird about this because I look at the proof given in the article, and I'm convinced of the statement about the partial derivatives of the two equaling each other. Though, I'm pretty sure it is actually the way that I have written it here. This is a much deeper statement. I think this should be changed :). Anybody interested in finding a source for this?
-- 24.7.197.142 ( talk) 08:49, 3 February 2011 (UTC)
Somewhere in the article it is written: "One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinatizations of the same symplectic manifold."
As an engineer I have no idea what a symplectic manifold is. It makes it rather difficult when one is trying to refresh one's ideas about Hamiltonians that one should now have to go and try and understand what "Symplectic manifolds are". Please - When writing articles proceed from the simple to the complex, from the concrete to the abstract and from the particular to the general and not vice versa Phillip ( talk) 06:04, 21 April 2011 (UTC)..
I'm and economist and I was looking for an economic interpretation for Hamiltonian. I didn't found anything useful here, but I did find a classic paper on the subject, " Introduction to Hamiltonian Dynamics in Economics". Do you think this should be a new section in this article? I think that hamiltonian function shouldn't redirect tho Hamiltonian mechanics, so other areas would only look for the use of Hamiltonian function. Best, Luizabpr ( talk) 17:19, 4 July 2011 (UTC)
I have changed the subsection Notes under Simplified overview of uses, tagged by Wikipedia:No original research:
I really do not think the current version adds anything helpful to the article. All that needs to be said (if anything) is:
This has replaced that section with no heading (no need) and the tag has been removed. Furthermore
-- F = q( E + v × B) 12:25, 16 February 2012 (UTC)
The section Geometry of Hamiltonian systems looks accurate enough, with proper links, but could surely be beefed up a little bit, perhaps with a diagram or figure with all the "ingredients". YohanN7 ( talk) 14:11, 5 October 2012 (UTC)
I don't really see the point of this section. That's a lot of evolved math for nothing but Hamiltonian systems (in the sense of physics). Maybe the symplectic point of view is already enough for the interested (mathematical) physicist. Just say, a Hamiltonian system can be considered as a symplectic manifold together with a smooth function $H$ on it. The manifold, which is usually the contangent bundle of a configuration space manifold $Q$, is symplectic, because the symplectic structure gives us a notion of poisson bracket and thus we can write down Hamilton's eq. of motion as a global flow equation (give that equation + explain). If needed, one can then go over to the time dependent case and set $\omega = d \theta - d E \wedge dt$ as well as $H_{new}= H_{old} - E$, where we take the cotangent bundle of $Q \times \R$. Also this "which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian." does not make sense. Yes, there is a natural symplectic structure (the cartan derivative of the canonical 1-form), but it is not a function, let alone the Hamiltonian. 94.211.47.103 ( talk) 20:44, 10 November 2013 (UTC)
I believe this article does not has, neither explain what is, the Hamiltonian mechanics' mathematical formalism. There is no section actually presenting the main mathematical premisses of Hamiltonian mechanics.
For example:
Another missing part in the article relates with the difference between Hamiltonian mechanics and classical mechanics...
I recently made a modification on the "overview of uses", which I believe fixes some of these problems. It was reverted based on an argument of "breaking < math >'s", which I believe is far from pragmatic (besides, every < math > was correct). I'm going to wait a bit until the user who reverted it to comment here, I'm sure the leading argument is not the < math >. In case of no response, I will revert my contribution back.
The introduction of this article is way too advanced for me. I am an educated amateur not a mathematician. How about an introduction ( or a separate article) along the style of "Hamiltonian Mechanics for Dummies". I got interested in HM from reading about Newton and I am curious about the trail from Newtonian mechanics onwards. My friends agree with me that many of the maths pages on Wiki have no suitable lead in section. Whoever thought of the idea "books for Dummies" was on to a great idea. — Preceding unsigned comment added by 118.208.122.190 ( talk) 22:23, 15 March 2014 (UTC)
I think that the Heisenberg Hamiltonian is confusing. If we take the coordinates $x,y,z$ like in the Heisenberg group page, it should be proportional to . I think the one that was meant in the article is the Hamiltonian in an non-holonomic basis, but in this case have nothing to do with the corresponding momenta for coordinates , which is some what confusing. — Preceding unsigned comment added by 147.122.44.52 ( talk) 14:55, 24 July 2018 (UTC)
Section Charged particle in an electromagnetic field says that the formulas are for Lorenz gauge. I believe the Lagrangian is valid for all gauges? Can someone confirm? Ponor ( talk) 14:43, 6 May 2020 (UTC)
2020-10-27 edit: this was discussing the 2020-05-06 version of the article, where the Hamiltonian was said to be valid in Lorenz gauge, as if it would be different in other gauges for the vector potential. This has been corrected since. Ponor ( talk) 12:41, 27 October 2020 (UTC)
@ Lantonov: In the section Hamiltonian mechanics#Deriving Hamilton's equations, it says "Since this calculation was done off-shell, one can associate corresponding terms from both sides of this equation to yield:". You added a "clarification needed" template to this. Off-shell simply means that the system may be following a path which is not the one required by classical physics, i.e. not the one which extremalizes the action. It does not have to have anything to do with quantum mechanics. JRSpriggs ( talk) 17:59, 26 October 2020 (UTC)
Another, smaller, problem (more like nit-picking) is that the Lagrangian in most popular textbooks and in the related Wiki articles, such as Lagrangian mechanics, Maupertuis's principle, Principle of least action, Calculus of variations, Hamilton's principle, etc., is denoted just while in this article it is denoted . In the same way, is more popular than (e.g. Schrödinger equation). Lantonov ( talk) 07:23, 27 October 2020 (UTC)
Another incongruence:
In the overview it is written that Hamilton's equations are
referring to a relative obscure textbook of Hand and Finch.
On the other hand when deriving Hamilton's equations later, they are
In the beginning we have two Hamilton's equations, and after derivation they become three. Which is true? I think that two is the right number, at least my textbook (Landau & Lifshitz) says so. Lantonov ( talk) 08:44, 27 October 2020 (UTC)
Relativity requires the more general form in which dependence on t is explicit because you must have "background" fields or constraining objects UNLESS you go whole-hog to a generally covariant theory of everything (which does not exist at present).
In other words, anything done using the Hamiltonian formalism must (for the time being) be an approximation based on non-relativistic assumptions.
If q0 = t, then p0 = − H.
JRSpriggs (
talk)
20:38, 28 October 2020 (UTC)
So called Energy based model are applied in machine learning, essentially usage of Hamiltonian mechanics. And redirect the page Energy based models to this subsection. -- mcyp ( talk) 18:39, 5 February 2022 (UTC)
I have created a new section to detail the case of which is taught in engineering and science courses. This follows on from the section Hamiltonian mechanics § Basic physical interpretation to give more precision and detail, including proofs and citations.
The section also somewhat relates to Lagrangian mechanics § Energy, which has been getting quite a few edits lately.
Are there are any comments about this new section, or anything that anyone would like me to address? DrBrandonJohns ( talk) 02:00, 22 February 2024 (UTC)