Talk:Hamiltonian mechanics

message: 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)
thread_title: Accessibility

message: 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)
thread_title: Intro

message: 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)
replies: text:  Good question. Hamiltonians are great for solving problems that involve transfer of energy, and momentum, between rigid bodies, and in some circumstances allow problems that would be hard to solve using just Newton to be solved in very few lines of equations. Examples include things like compound pendulums and so on. However, very few engineers use Hamiltonians in their day to day lives, to a large extent they have been replaced by kinematic programs like MSC ADAMS, which have the advantage of being able to easily handle flexible bodies, non ideal contact with damping, and friction. I did not study them at uni, and must confess that this page is absolutely no help at all to me, and I work in the area of dynamics. I'm sure it makes sense to physicists. Greglocock 04:48, 3 January 2007 (UTC)
thread_title: Page move

message: 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)
replies: text: \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)
thread_title: Norm

message: 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)
replies: text: 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)
replies: text:  the Hamiltonian system page already exists. This problem is solved. Jorgecarleitao (talk) 14:21, 5 March 2013 (UTC)
thread_title:  Hamiltonian systems 

message: The result was merge sbandrews (t) 18:51, 23 June 2007 (UTC)
replies: text: That sounds like a good idea, and I'd support the move. — Rebelguys2 talk 03:54, 11 April 2007 (UTC)
replies: text: make sense, I'll have a go... sbandrews (t) 10:36, 10 June 2007 (UTC)

text: ok I'm done, comments or reverts welcome... sbandrews (t) 11:00, 10 June 2007 (UTC)
thread_title:  Proposed mergefrom Hamilton's equations 

message: 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)
replies: text: No? I shall remove it then. AstroMark (talk) 14:03, 27 August 2008 (UTC)
thread_title:  Mnemotechnics 

message: 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:!jimtalk  contribs 22:45, 13 December 2008 (UTC)
thread_title:  Example? 

message: 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)
thread_title:  section on deformable bodies 

message: 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)
thread_title:  derivation by time 

message: Am I blind or isn't there any article about classical hamiltonian itself (a function)?
Magdulewicz (talk) 13:49, 23 June 2010 (UTC)
replies: text: I think that the function (in classical physics) is adequately covered in this article. Is it not? What else would you have us say about it which is not already in this article? JRSpriggs (talk) 06:39, 24 June 2010 (UTC)
replies: text: I guess I'd talk about effective potentials, but that's just my two cents. Effective potentials using the hamiltonian are an interesting thing that you can do without talking about the dynamics of the situation.--24.7.197.142 (talk) 08:53, 3 February 2011 (UTC)
thread_title:  Article about hamiltonian 

message: 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)
replies: text: According to Marsden & Ratiu (1999) p. 231 the notation T and V seems to go back (at least) to Lagrange. In a footnote they refer to further possible sources on historical background. -- Crowsnest (talk) 06:45, 14 January 2011 (UTC)
replies: text: Merci, mais je reste avec la même question, pourquoi V? <<un champ Vectoriel?>>. Thank you for looking into this, but I didn't see what V stands for. Perhaps "champ Vectoriel"?Laburke (talk) 20:22, 13 March 2011 (UTC)
replies: text: Frankly at this point in time, it does not matter for what word "V" is short, if indeed there is any such word. Given the immense multitude of concepts and the limited number of mathematical symbols available to represent them, re-use of symbols (overloading) and arbitrary choice of symbols (merely because they are different from some other symbols one has chosen to use) are bound to occur. JRSpriggs (talk) 02:22, 14 March 2011 (UTC)
thread_title:  Symbols 

message: 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)
thread_title: Approach

message: 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)
replies: text: I'm now sure what an economics interpretation of the Hamiltonian would be. This is an article about Hamiltonian mechanics, which is a branch of physics and not related to economics in any way. About the Hamiltonian function, it is the same thing as the Hamiltonian, and hence it has to redirect here (or if Hamiltonian function can mean more things, a disambiguation page should be created). —Kri (talk) 19:32, 13 August 2011 (UTC)
replies: text: Many of the top economists today have advanced degrees in physics. These economist are able to apply techniques of mathematical physics in work on economics. Reviewing this paper it is an application of the technique - not even a generalization and should be included here to illustrate that Hamiltonian techniques can be used in other disciplines. Some pure matematical uses are mentioned in  http://terrytao.files.wordpress.com/2008/03/hamiltonian.pdf OrenBochman (talk) 09:44, 11 February 2012 (UTC)

text: The Hamiltonian is pretty basic to dynamic models in economics these days. I'm surprised that six years after these comments there is still no mention of it (or even a link or disambiguation) here. I would encouage someone to add something helpful. Winter84 (talk) 17:35, 16 August 2018 (UTC)
replies: text:  Read Hamiltonian (control theory) and Hamiltonian system. Lantonov (talk) 10:21, 27 October 2020 (UTC)
thread_title:  Hamiltonian for economics 

message: I have changed the subsection Notes under Simplified overview of uses, tagged by Wikipedia:No original research:
* "Hamilton's equations are appealing in view of their beautiful simplicity and (slightly broken) symmetry." <--- Are they not symmetric in \pm q \rightleftharpoons \mp p and \pm \dot{q} \rightleftharpoons \mp\dot{p}? A little too much hype?
* "They have been analysed under almost every imaginable perspective, from basic physics up to symplectic geometry." What does "basic physics" mean? A simple physical system with one/very few degrees of freedom? Elementary physics at school level? I can't think of anything.
* "A lot is known about solutions of these equations, yet the exact general case solution of the equations of motion cannot be given explicitly for a system of more than two massive point particles." <--- The sentence is sort of fine but partially redundant. Generally solutions to differential equations can be analysed qualitatively and quantitatively, with approximations and in limiting cases. They can’t always be found explicitly. 
* "The finding of conserved quantities plays an important role in the search for solutions or information about their nature." <--- This is randomly stuck in, and its easy to read conservation laws from the lagrangian (finding cylic coords). The Hamiltonian itself is conserved for time-independent Lagrangian functions.
* "In models with an infinite number of degrees of freedom, this is of course even more complicated." <--- Fine, but redundant. Why talk about infinite degrees of freedom when a finite number is easier to comprehend? Also arn't physical models supposed to simplify the siuation and have as few degrees of freedom as possible? Not always, but usually.
* "An interesting and promising area of research is the study of integrable systems, where an infinite number of independent conserved quantities can be constructed." <--- Source? Why is it so helpful to have an infinite number of conserved quantities? What will a reader get from this?
I really do not think the current version adds anything helpful to the article. All that needs to be said (if anything) is:
* "Hamilton's equations are symmetric in \pm q,\mp p, and hence \pm \dot{q},\mp\dot{p}. Naturally, the more degrees of freedom the system has, the more complicated its behaviour (predicted by the solutions), since the degrees of freedom correspond to the configuration of the system i.e. (generalized) positions, momenta and the rates at which these change (time derivatives). As such, for more than two massive particles the solutions cannot be found exactly - the many-body problem. It is still possible to obtain qualitative knowledge about the system by approximative analysis of the differential equations."
This has replaced that section with no heading (no need) and the tag has been removed. Furthermore
* "Generally, these equations do not provide a more convenient way of solving a particular problem in classical mechanics. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science." <--- Are the equations not convenient due to their symmetry and that they are 1st order differential equations? There is no problem in using more than one formulation of mechanics, so this has a slightly negative impression on the equations. I changed it to be more neutral;
* "Sometimes, but not always, these equations provide a convenient way of solving a particular problem in classical mechanics. Also, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science."
--  F = q(E + v × B) 12:25, 16 February 2012 (UTC)
replies: text: Also I re-wrote the instructive/textbooky written section the using H equations section. WP is not supposed to be written like this (WP is NOT...). More edits of this nature can probably be done but I'm too busy for now. --  F = q(E + v × B) 12:41, 16 February 2012 (UTC)
replies: text: Forget the crossed out content above - I stated the obvious and it was unhelpful/wrong, so the paragraph in the lead will be re-phrased to the original wording. =( -- F=q(E+v×B) ⇄ ∑ici 10:07, 18 March 2012 (UTC)
thread_title: Re-write section of "beauty", also a pinch negative

message: 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)
thread_title:  Geometry of Hamiltonian systems 

message: 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:
# Hamiltonian mechanics requires canonical coordinates variables and not just "generalized coordinates". This is not introduced.
# the fact that one of the generalized variables is attached to the reference frame chosen is also not explained, which leads to a big confusion of the article... The "momentum" is not arbitrary, it is fixed by the canonical conjugate relations...
# While Hamiltonian was historically connected by the Lagrangian mechanics, this is no longer true. This can be supported by different sources, such as the chapter on Hamiltonian system of Edward Ott's book "Chaos in Dynamical Systems".
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.
:Using TeX for inline equations is not popular. If you use HTML markup for the inline equations, then focus might be more on the content part of your changes. YohanN7 (talk) 19:55, 7 March 2013 (UTC)
thread_title:  What is Hamiltonian mechanics mathematical formalism 

message: 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)
thread_title:  Please simplify the introduction 

message: 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 (p_x -p_z y/2)^2 + (p_y + p_z x/2)^2. I think the one that was meant in the article is the Hamiltonian in an non-holonomic basis, but in this case p_x,p_y have nothing to do with the corresponding momenta for coordinates x,y, which is some what confusing.  — Preceding unsigned comment added by 147.122.44.52 (talk) 14:55, 24 July 2018 (UTC)
thread_title:  Sub-Riemannian geometry example 

message: 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)
replies: text: Lagrangian and Hamiltonian are valid for all gauges (measurement systems). That's why they are defined in generalized coordinates which do not depend on which coordinate system or gauge is used. In the physical jargon it's called "gauge-invariant". It's just that the particular formulas that are used in that section are valid for Lorenz gauge. Those formulas would be different if another gauge is used. Lantonov (talk) 10:36, 27 October 2020 (UTC)
replies: text: Hi  I would not confuse gauge invariance with the choice of generalized coordinates. My concern was only about the minimal substitution part. At that time, the article said the Hamiltonian Lagrangian  \mathcal{L} = \sum_i \tfrac{1}{2} m \dot{x}_i^2 + \sum_i q \dot{x}_i A_i - q \varphi was (only?) valid in Lorenz gauge, while it's actually valid in any: take As and φs in whichever gauge you like, put them in, and physics will be the same. Ponor (talk) 12:41, 27 October 2020 (UTC)
replies: text: You mean Lagrangian instead Hamiltonian. Well, at least now the article does not claim that the Lagrangian is valid only for the Lorenz gauge. The formula is also different replacing the first term with its relativistic counterpart (Lorenz transformation). Since the classic Lagrangian is a generalisation for the Newtonian mechanics (low velocities), for relativistic mechanics the expressions should be transformed appropriately for the Langrangian to be valid there too. Lagrangian is also valid in general relativity, that is, in curved space, with still more modifications and transformations like Levi-Chivita connections, covariant derivatives; metric, Riemann, and Ricci tensors, and so on. Lagrangian didn't fit well in quantum mechanics but Hamiltonian did. The problem with the Lagrangian is that it subsumes a preservation of mass. For example, \partial L / \partial \dot{\mathbf{q}} is equivalent to the Newtonian momentum \mathbf{p} = m \mathbf{v} where only the the velocity \mathbf{v} is supposed to be changing. In relativity and quantum physics, this is no longer true: mass may become energy and vice versa. However, the energy conservation laws are valid (at least for special relativity and quantum mechanics) so the Hamiltonian as the expression of the total energy of the system should be preserved there. Lantonov (talk) 13:33, 27 October 2020 (UTC)
replies: text: (Yes, I meant Lagrangian, sorry about that, fixing it now) And yes, there was a discussion about the issue elsewhere (another editor's user page) and it got fixed. I've never seen these relativistic L and H used in any real problems, and I'm only talking about the nonrelativistic ones here. While the Lagrangian may not be relativistically invariant, you still have the liberty of choosing the gauge for your As and φs. Normally you'd be told what E and B are (because they can be measured) and you'd need to know which gauge to use for the potentials in the expressions for L and H. Ponor (talk) 14:04, 27 October 2020 (UTC)
thread_title:  Charged particle in an electromagnetic field 

message: 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 L while in this article it is denoted \mathcal{L}. In the same way, H is more popular than \mathcal{H} (e.g. Schrödinger equation). Lantonov (talk) 07:23, 27 October 2020 (UTC)
thread_title:  Notation 

message: 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)
thread_title:  A new subsection Applications in machine learning

message: I have created a new section to detail the case of
\mathcal{H} = T + V
which is taught in engineering and science courses. This follows on from the section  to give more precision and detail, including proofs and citations.
The section also somewhat relates to , 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)
thread_title:  New section on Hamiltonian as the total system energy