Talk:Hamilton–Jacobi equation/Archive 1

Archive 1

Level of article

(copied from User talk:Linas by Oleg Alexandrov (talk))

I reviewed the article on HJE and canonical transformations. Looks good, although we have a systemic problem: everything you wrore is from the perspective of an undergrad physics/engineering major. However, the best way of understanding what is "really going on" is by means of geometry, in the language of manifolds. I'm not sure how to resolve this tension; both perspectives are needed. linas 17:14, 4 June 2006 (UTC)

Hi, Linas, thanks for the quick feedback! I agree that Wikipedia would benefit from having both perspectives on the HJE, but I'm concerned that they might not both fit well into one article. For example, some readers might not want the full geometrical description in terms of manifolds, since the concepts would be unfamiliar and understanding the article would likely require more effort than they could easily invest. I confess, I find concepts like cotangent bundle a little scary, although I'm sure they'd be clear if I spent more time trying to understand them. Perhaps we should have two articles, Hamilton-Jacobi equations (physics) and Hamilton-Jacobi equations (mathematics), and a disambiguation page that clarifies the differences between them? That might give enough room to have two cogent articles at different levels, without trying to do everything in one article. What do you think? I tried something simiar with canonical transformation vs. symplectomorphism. WillowW 17:54, 4 June 2006 (UTC)

Hmm, I disagree. First, re the titles: even physicists use the language of manifolds now; so the distinction math/physics is false. The folks writing the textbooks are employed by physics depts. I was tempted to say that only engineers stick to the rather dry Euclidean form, but even that's not true: I've seen books on robotics that launch into algebraic varieties on page one, and holonomy by page 20 or 30. Work on both satellite motion, and space-craft inter-planetary travel also uses the modern language; I am hard-pressed to think of an application in physics or engineering that doesn't use the modern language. No, it would be a dis-service to split in this way. A better split might be to devote a single article page to each separate example. linas 18:14, 4 June 2006 (UTC)

Hi, Linas, I can see why it's good to keep the article together. However, I feel that we have to keep the initial part of the article intelligible to people who have learned only multi-dimensional calculus. Otherwise, we're likely to lose >98% of our readers, since most scientists and even lay people have learned calculus but very few have studied Riemannian geometry or manifolds/cotangent bundles/algebraic varieties/etc. According to the Science Citation Index from 1980-2006, there were 2284 articles about the Hamilton-Jacobi equation; of these, only 1 (!) mentioned "tangent bundle", "symplectic form" or "holonomy" in their title, keywords or abstract; exact zero of the HJE articles mentioned "symplectomorphism" or "algebraic variety" in the same places. "Manifold" and "geodesic" fared a little better, with 54 (~2.4%) and 35 (1.5%) articles, respectively. These data suggest that >98% of scientists are using the HJE in its old-fashioned calculus-based form. So I suggest that we include the more sophisticated, modern topics at the end of the article -- do you agree? WillowW 23:51, 4 June 2006 (UTC)

I think you severly underestimate the intelligence of authors. I doubt anyone who publishes an article today on the HJE would not have had a good grounding in parital differential equations, and it is impossible to study PDE's without learning a good bit of geometry. Tangent bundles are not exactly complicated; this is part of the undergraduate math curiculum, with the bare basics coming at the sophomore level. All math majors and most physics majors will have at least the basic concepts down. What edits are you proposing? linas 00:10, 5 June 2006 (UTC)

I agree with Willow here. It never hurts to keep things accessble. Besides, we don't want to address it to people who publish on HJE, rather, to people who want to learn about it. See also Wikipedia:Make technical articles accessible. Oleg Alexandrov (talk) 01:28, 5 June 2006 (UTC)

Aww, com'on. You know me better than that. Anyway, perhaps we should move this to the talk page of that article. The only proposal that I'd have is to promote the various examples to thier own WP articles, and leave the main article to talk about generalities, instead of diving into specifics.

linas 02:56, 5 June 2006 (UTC)

Hello, I am reasonably aquainted with these concepts,however keeping this article simple as it is done here would be very good as I could clear many of my doubts.Had the article been more techincal,It would have frightened me like many textbooks do.This article is excellant, please don't bring in any modification to satisfy people who want more rigour. regards Sathya

The intro is useless

Could people please write intro paragraphs in simple language? I have an undergrad physics background, and I still don't understand most of the terminology used. I somehow doubt there's any actual physics in this page that can't be reduced to a simple pure-English statement, as opposed to a jargon-loaded mouthful that links to other equally illegible articles. If you'd like to see a counterexample (well, almost anyway), consider Lagrangian mechanics, which clearly and simply explains what the heck it actually IS.

I might not have the physics exactly right, but if I am reducing it correctly, the intro should read something like "In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a mathematical technique that produces a single equation that describes dynamic behavior of a mechanical system. Traditional methods in classical mechanics, like Hamilton's equations of motion, generally develop a set of related equations that must be solved, whereas the HJE is notable in that it produces a single equation containing the same information."

Is this even close to the truth? If so, why doesn't it look like that? If not, then how does one expect anyone to understand it?

Maury 21:36, 31 July 2006 (UTC)

Dear Maury,

I'm sorry that you don't find the introduction intelligible, and I'll be glad to try and fix it up. I'm still a relative newbie, so it's hard to know the proper level at which to write such technical articles. On the one hand, some users are annoyed that I made the article as basic as it is, (e.g., using an undergraduate-calculus presentation) whereas others like yourself are mystified from sentence #1 onwards.

Usually, my personal style is to "lay out a honey trail to enlightenment", i.e., to start simply and gradually work up to more sophisticated results. I do feel, however, that we shouldn't omit the more sophisticated results just because 90% of Wikipedia's readers won't have the background to understand them; we owe it to the remaining 10% to do justice to the whole topic. Admittedly, I didn't follow the "honey-trail" approach here; maybe I was under pressure? Whatever the reason, I'll try to improve it now and I look forward to your future suggestions for this and other articles.

I'm away from home now, nursing a family member, so I may not be able to reply or work on the articles regularly over the next few weeks, but I'll do my best. Willow 03:50, 1 August 2006 (UTC)

Much better!. This is precisely the sort of intro I think makes these articles readable. Even someone who isn't so interested in the math itself can now understand why the math is important. However, I am still curious if the "single" means what I think it does, that the system reduces normal approaches (like the Lagrangian) to a single equasion. If so, I think that too deserves mention in the intro. Maury 12:47, 11 August 2006 (UTC)

Inequivalence of mechanics methods

For the most part, the intro is good, but there is one issue that needs to be fixed: neither the Hamilton-Jacobi equation, nor the Lagrangian or Hamiltonian approaches are equivalent to Newton's Laws of Physics. For one, the former three approaches transcend theory-boundaries and apply just as well to Relativity and (indeed) serve as a means of generating new theories from scratch. But, just as importantly, the inequivalence goes the other way too: not all systems described within Newtonian physics are Lagrangian, Hamiltonian or have a Hamilton-Jacobi equation. The question of when a physics described by a second order law of motion (such as the law of force in Newtonian physics) has a Lagrangian is the inverse Lagrangian problem, and the answer is a set of conditions known as the Helmholz conditions. Then you have the issue of the equivalence of Hamilton-Jacobi vs. Hamiltonian or Lagrangian systems, which is not generally true either. The three approaches are closely related, but not all equivalent. Mark (129.89.32.117) 23:44, 18 November 2006 (UTC)

Problem with variable separation for constant Hamiltonians

There is a statement in the article that in case H doesn't depend explicitly on time, "the time derivative ${\frac {\partial S}{\partial t}}$ in the HJE must be a constant." The word must is wrong. Consider the Hamiltonian for the free particle $H=p^{2}/2m$. One possible solution of the corresponding equation is

$S(q,t)={\frac {mq^{2}}{2t}}+const,$

which does not have a constant time derivative.

What is true however, is that there exists a solution which has the form $S=W(q_{1},\dots ,q_{N})-Et$, provided the Hamiltonian is constant. This statement is not trivial to prove, as far as I can say. --Avatariks 16:55, 28 February 2007 (UTC)

Errata

There was a mistake in the first equation,

$H\left(q_{1},\dots ,q_{N};{\frac {\partial S}{\partial q_{1}}},\dots ,{\frac {\partial S}{\partial q_{N}}};t\right)-{\frac {\partial S}{\partial t}}=0,$

which has been changed to

$H\left(q_{1},\dots ,q_{N};{\frac {\partial S}{\partial q_{1}}},\dots ,{\frac {\partial S}{\partial q_{N}}};t\right)+{\frac {\partial S}{\partial t}}=0.$

This was done to be consistent with a solution of the form

$S=W(q_{i})-Et,$

which is used later in the article. The notational adjustment conforms to standards used in classical dynamics.

Best,

-Js

158.144.51.79 07:08, 10 November 2007 (UTC)

Comments

WillowW: You asked for my comments; here they are: A quick skim shows that the page is much expanded; that's good. I note, however, that it limits itself to Euclidean space; there should be some rome in the artcile for stating that the equaitions are far more general than that, and find broad applicability. For example, the geodesic flow is just Hamiltonian flow: By using HJE to solve the Hamiltonian $H=g_{ij}p^{i}p^{j}$ where g is an arbitrary smooth metric tensor, one gets the equaitons of motion for a geodesic; in fact, *all* geodesics in Riemannian geometry (and the various specializations thereof) are obtained by HJE. This is the most important example of the importance of HJE to geometry in general, rather than just classical mechanics. linas 15:32, 4 June 2006 (UTC)

I added a bit about the geodesics. Unfortunately, its a rough fit for the article as it currently stands. There are numerous problems:

• A better structuring of the article should be provided, so that the elementary examples, in Euclidean geometry can be presented, without cluttering the fact that the HJE are studied in much broader and deeper contexts than merely undergrad physics.
• Somehow, it should be emphasized that the solution and structure of HJE is best understood in terms of modern notions of geometry, and symplectic geometry in particular.
• The equivalence of the Euler-Lagrange and the HJE approach should be clearly noted.
• Examples of mechanical systems that are solved by Riemannian manifolds shoulld be given. I can think of two, the Artin billiards and the Hadamard billiards, both examples from dynamical billiards.
• Existance solutions, and thier completeness, should be discussed. Hopf-Rinow theorem, the Chow-Rashevskii theorem. That Morse theory tells you about the structure of the solutions should be mentioned. For example, in $E=n\hbar \omega +1/2$ for the simple harmonic oscillator, the the 1/2 arises because it is twice the Morse index (known as Maslov index in physics).

linas 16:55, 4 June 2006 (UTC)

What's with all the different coordinate systems (eg. varieties of cylindrical)?? Aren't they irrelevant distractions to the actual topic of the article? Does every single mathematical physics article need to have sections demonstrating how an equation can be transformed into five arbitrary coordinate systems? It seems like a trivial (albeit tedious) exercise; wikipedia would be better served by shorter articles (with more specific and important examples), plus a separate general article on coordinate transformations. If I'm missing something, could the importance of the present sections be explained? 124.168.196.2 02:17, 1 December 2007 (UTC)

Historical references

I'm very interested in learning more about some of the history that's only obliquely referenced here (and in other articles.) For instance, at the beginning of the article, it is said "The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle." What was Johann Bernoulli's view of this? What did his contemporaries think? As an undergraduate physics major who has a copy of Goldstein and has seen and followed its derivation of HJE, and its analogy with Schrodinger's equation, I want to see where this all came from. The article on Schrodinger's equation also mentions that he was driven by HJE, and that he "secluded himself in a mountain cabin with a lover" when he thought he had failed. These are tantalizing tidbits. Can anyone provide any references to further reading? Amplimax (talk) 02:38, 12 July 2008 (UTC)

Optimization

Isn't the Hamilton-Jacobi equation just a necessary condition for optimality in a certain class of calculus-of-variations problems? I thought it also had a lot of applications besides physics. —Preceding unsigned comment added by 64.178.105.25 (talk) 05:11, 10 October 2008 (UTC)

Fathi?

I think Albert Fathi's book Weak KAM Theorem in Lagrangian Mechanics (in preparation; several versions available online), as it expounds on the theory related to the viscosity solutions of the Hamilton-Jacobi equation. 140.180.171.121 (talk) 18:39, 10 November 2009 (UTC)

HJE $S$ vs. action

In "Mathematical formulation", the derivation of the formula for the conjugate momenta starts from expressions for $\delta S$, which are taken from the definition of action. The connection to the $S$ that appears in the HJE is certainly not trivial. The fact that $S$ equals the action is only mentioned later, in "comparison with other formulations" (and even there it is described as "remarkable"). This makes the "formulation" section very confusing unless you have prior knowledge of the topic--AmitAronovitch (talk) 09:16, 24 June 2010 (UTC)

Relations to Schrodinger's wave equation

There is a citation request at the beginning of the article. One relevant citation is: J.J. Sakurai, Modern Quantum Mechanics, Chapter 2.4 Schrodinger's wave equation - The classical limit.85.65.127.0 (talk) 02:43, 5 July 2010 (UTC)

Thank you for the suggestion! I've been meaning to come back here and reference this article for years, but I've always gotten distracted. I provided the reference you suggested, and supported it with one from Goldstein. Willow (talk) 05:25, 5 July 2010 (UTC)

Towards the end of the article there is misleading substituition in Schrodinger's wave equation. A correct substitution is equation 2.4.25 in Sakurai book. Unfortunately I don't know to edit math equations or references in Wikipedia.85.65.127.0 (talk) 04:40, 5 July 2010 (UTC)

Thank you, and it's perfectly fine if you don't want to edit the equations, which are written in LaTeX; I or others will be happy to do that for you. However, the goal of this section was only to present the parallelism between classical mechanics and the classical limit of Schrodinger's equation. I believe that the present derivation might be better for this goal, having only one non-classical term - what do you think? IMHO we need not present the full Schrodinger equation, although the equation you cite is certainly more accurate in allowing ψ₀ to vary with time and position. Unfortunately, it is also more complex, with several terms not found in the classical limit. It is also expressed in terms of the probability density ρ, rather than the probability amplitude ψ, which might confuse non-experts. Perhaps we can marry both approaches by adding the Sakurai equation after the derivation and show the relation to the WKB approximation? There's certainly a lot left to add to this article! :) Willow (talk) 05:25, 5 July 2010 (UTC)

What is $ S $?

This needs to be defined carefully. In particular, how does $ S $ depend on $ \alpha $? This article is effectively unreadable. — Preceding unsigned comment added by 108.74.160.24 (talk) 03:44, 19 June 2016 (UTC)

Schrodinger equation

The article says that

${\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -U\psi ={\frac {\hbar }{i}}{\frac {\partial \psi }{\partial t}}$

is a nonlinear Schrodinger equation. In fact, this is exactly the standard Schrodinger equation (with a minus sign on both sides because 1/i = -i). And regardless, everything acting on ψ is a linear operator, so it's not a nonlinear differential equation.

In the next part, where we substitute our ansatz for ψ into the actual Schrodinger equation it says

${\frac {1}{2m}}\left(\nabla S\right)^{2}+U+{\frac {\partial S}{\partial t}}={\frac {i\hbar }{2m}}\nabla ^{2}S.$

This expression seems to have lost two minus signs and a ψ. Additionally, it seems to me that the terms are arranged in a confusing way, so it's not obvious where they come from. Might it not be more clear to write it as

${\frac {1}{2m}}\left[i\hbar \nabla ^{2}S+\left(\nabla S\right)^{2}\right]\psi +U\psi ={\frac {\partial S}{\partial t}}\psi$

Notice the term with the Laplacian of S has a different sign when moved to the right-hand side, although this doesn't affect the classical limit, because it goes to zero either way as ħ goes to zero. The time derivative has picked up a minus sign as well and doesn't carry a factor of ħ, so it would result in a minus sign on that term in the proceeding equation. Mpalenik (talk) 17:27, 17 January 2020 (UTC)

I completely agree that this is simply the (linear) Schrodinger equation. However, I think the article gets the signs correct. Remember that

$\psi =\psi _{0}\exp \left({\frac {i}{\hbar }}S\right)$

and so

$\nabla \psi =\psi _{0}{\frac {i}{\hbar }}\nabla S\exp \left({\frac {i}{\hbar }}S\right)$

$\nabla ^{2}\psi =\psi _{0}\left[{\frac {i}{\hbar }}\nabla ^{2}S+{\frac {i}{\hbar }}\nabla S\cdot {\frac {i}{\hbar }}\nabla S\right]\exp \left({\frac {i}{\hbar }}S\right)$

and so we get after dividing out the common ψ

${\frac {\hbar ^{2}}{2m}}\left({\frac {i}{\hbar }}\nabla ^{2}S-{\frac {1}{\hbar ^{2}}}\nabla S\cdot \nabla S\right)-U={\frac {\partial S}{\partial t}}$

or

${\frac {i\hbar }{2m}}\nabla ^{2}S-{\frac {1}{2m}}\nabla S\cdot \nabla S-U={\frac {\partial S}{\partial t}}$

which is what the article states. The form in the article is probably to emphasize that this is an eikonal equation.

Katachresis (talk) 21:33, 18 February 2020 (UTC)

Attention needed in the "Hamilton's principal function" def section

This part doesn't make sense. Particularly, that we're going nxn Hessian matrix to inverted matrix which I have never heard of to a vector equality. This seriously needs to be elaboration.

Let the Hessian matrix ${\textstyle H_{\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)=\left\{\partial ^{2}{\cal {L}}/\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}\right\}_{ij}}$ be invertible. The relation

${\frac {d}{dt}}{\frac {\partial {\cal {L}}}{\partial {\dot {q}}^{i}}}=\sum _{j=1}^{n}\left({\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}}}{\ddot {q}}^{j}+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {q}^{j}}}{\dot {q}}^{j}\right)+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial t}},\qquad i=1,\ldots ,n,$

shows that the Euler–Lagrange equations form a $n\times n$ system of second-order ordinary differential equations. Inverting the matrix $H_{\cal {L}}$ transforms this system into

${\ddot {q}}^{i}=F_{i}(\mathbf {q} ,\mathbf {\dot {q}} ,t),\ i=1,\ldots ,n.$

Ehaarer (talk) 03:21, 27 January 2023 (UTC)

Another particular issue here is that the notation ${\textstyle {\cal {L}}}$ is pulled out of nowhere. A line could easily be added to the "Notation" section about this. Mosher (talk) 17:24, 31 March 2023 (UTC)

Removing unreferenced claim in lead section.

I'm removing this sentence:

The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely[citation needed].

I did not find anything in Goldstein to support this particularly useful claim, despite his extensive treatment of both conservation and Hamilton-Jacobi equation. Nothing online either.

The sentence in the lead does not summarize content in later in the article. This is the only mention.

The claim may be true, but I don't think an unsupported but not obviously true sentence belongs in the lead.

I would support reverting this removal with an appropriate reference. Johnjbarton (talk) 00:52, 11 June 2023 (UTC)

Relationship to the Schrodinger equation

In this section it is claimed that after using the ansatz $\psi =\psi _{0}\exp(iS/\hbar )$ the Hamilton-Jacobi equation becomes the Schrodinger equation. But how does the non-linear piece $(\nabla S)^{2}$ coming from $p^{2}\to (\partial S/\partial q)^{2}$, and eventually from $H=p^{2}/(2m)+V(x)$, become the linear $\nabla ^{2}\psi$? Arlrequin (talk) 05:29, 4 July 2023 (UTC)

In the case that ψ=ψ₀exp(iS/ħ), then grad(ψ)=i*grad(S)ψ/ħ, and the gradient of that gives you 2 terms. With a bit of algebra you arrive at the equation in which the RHS is proportional to ħ. And when ħ goes to 0, you get the result. Galactic cosmicray (talk) 02:09, 11 September 2023 (UTC)

I believe the question refers to the first paragraph and the phrase "The Hamilton–Jacobi equation is then rewritten ...". The Hamilton-Jacobi equation is for S and the article content does not make it clear or nor reference how to manipulate the equation for S into an equation for $\psi$ as claimed. Johnjbarton (talk) 14:58, 11 September 2023 (UTC)

Possible mistake in the derivation of the formula ${\frac {\partial S}{\partial q^{i}}}=\left.{\frac {\partial {\cal {L}}}{\partial {\dot {q}}^{i}}}\right|_{\mathbf {\dot {q}} =\mathbf {v} }$

In step 1 of the proof of this formula it is stated that « If $\xi$'s starting point $\mathbf {q} _{0}$ is fixed, then, by the same logic that was used to derive the Euler–Lagrange equations, $\delta \xi (t_{0})=0$. » But i do not think this holds, the path $\xi$ has fixed start and end points, but if the perturbation $\delta \xi$ is assumed arbitrary as in the text then we have in general $$\delta {\cal {S}}_{\delta \xi }[\xi ,t;t_{0}]={\frac {\partial {\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} )}{\partial \mathbf {\dot {q}} }}\delta \xi (t)-{\frac {\partial {\cal {L}}(\mathbf {q_{0}} ,\mathbf {{\dot {q}}_{0}} )}{\partial \mathbf {{\dot {q}}_{0}} }}\delta \xi (t_{0}).$$ and the second term does not necessarily vanish, as we may have $\delta \xi (t_{0})\neq 0$ -and ${\frac {\partial {\cal {L}}(\mathbf {q_{0}} ,\mathbf {{\dot {q}}_{0}} )}{\partial \mathbf {{\dot {q}}_{0}} }}\neq 0$. We must explicitly assume that $\delta \xi (t_{0})=0$, that is that the perturbation does not alter the path's origin $\xi (t_{0})$. This is what is done in step 2 where one works only with what is called a "compatible" variation of the path, this includes the condition $\gamma _{\epsilon }(t_{0})=\gamma (t_{0})$ for all $\epsilon$, which implies $\delta \xi (t_{0})=0$. I think we can adapt the proof to allow moving origin, thus general perturbations, but we must also adapt the definition of "compatible" -drop the assumption at the origin.

A few remarks on the literature: fixed origin for perturbations is also assumed in Arnold's book on classical mechanics, though his proof does not use Gateaux derivatives but works only with integrals. Abraham and Marsden's Foundations give a rather fancy proof, which yields a more general result -including moving origin, if im not mistaken. Evans in his book on PDE uses the Hopf-Lax formula for a minimizer of the action, he proves that functions satisfying it, for a convex lagrangian, satisfy the HJ equation where it is differentiable -the action functional is then only Lipschitz thus only almost everywhere differentiable. Physics text that i found offer only very sketchy proofs. Plm203 (talk) 13:38, 12 November 2023 (UTC)

Two different definitions of Hamilton's principal function in this article.

In the section "Hamilton's principal function" it is defined as the action (consistent with Feynman). In that definition it is a definite integral and defined by that integral.

In the section "Action and Hamilton's functions" the principal function is defined consistent with Goldstein v3 as the indefinite integral resulting solutions to the H-J equations.

If you read Feynman's logic, the first definition is wrong. He says no one wants to write out "Hamilton's principal function" all the time, let's all it "action". Then Goldstein still gets the Hamilton's principal function name for his version.

Unless some objects I will make that change next time I come around here. Johnjbarton (talk) 02:20, 17 November 2023 (UTC)