Talk:Symplectic integrator

A question

It says in the article:

"Symplectic integrators are designed for the numerical solution of Hamilton's equations"

What happens if you try to use a symplectic integrator whose time evolution cannot be derived from a hamiltonian (eg. if the system also includes a dissipative force)? I think this should be mentioned in the article. Ossi 12:48, 25 October 2007 (UTC)

A symplectic integrator is defined in terms of a Hamiltonian, H. If there is no H (as you say, the system cannot be derived from a Hamiltonian) then there is simply no integrator to iterate. Does that make sense? Jcandy 22:16, 27 October 2007 (UTC)

An other question

This article is supposed to be of "mid-importance within physics" however it is written with "advanced" mathematic notations. As a physician, I can not understand it and I find this quite unfortunate. Conversely it is perfeclty easy to read and implement algorithm as describe in page like Runge-Kutta_methods. I must admit that the form used to describe the algorithm here is elegant and short. It is quite likely the prefered form for mathematicians. But in order to broaden the audience I would ask anyone reading this comment and being able to translate this article into more traditional and ancient algorithm language to perform this translation. Assuming such a translation is possible. If not I beg for a short explanation of the equation $z=(p,q)$ it hinders my understanding of those lines. regards. 192.93.101.133 (talk) 07:16, 24 September 2008 (UTC)

In the text, $z$ is defined to name the set of position coordinate q and momentum coordinate p. For example, if you are thinking about the motion of a particle along a coordinate $x$ and if its momentum is $p_{x}$, then $z$ just means a pair of variables, $(x,p_{x})$. If you are thinking about a particle in a 3D space and representing its position by the coordinates $(x,y,z)$ and its momentum by $(p_{x},p_{y},p_{z})$, then $z=(x,y,z,p_{x},p_{y},p_{z})$. The symbol $z$ was introduced only to simplify the expression so that the essense of the derivation become clear. I added an explanation in the text. Norio —Preceding undated comment added 05:54, 1 August 2010 (UTC).

Its quite a few years since the above question was asked. I think this article is one of many I have seen which are less comprehensible than they should be. In the spirit of Wikipedia have decided to be bold and clean up this one. I am first going to insert some personal insights that motivate what I want to accomplish and which address the concerns expressed in the original question.

I have only been examining this particular Wikipedia article since somewhat over a month ago. In my humble -- but I think widely shared -- perception it is an example, though a relatively mild one, of a plague that has descended upon articles at this level throughout Wikipedia. I don't want to start a discussion on the plague as such at this time. As it is, what I will say about it may well inspire well meaning devotees to characterize my remarks as demonstrating too much Point of View. I'll take that risk for three reasons: (1) I am a PhD physicist and so I have a POV that is more professional than personal,(2) the problem I am going to address is a growing quality and policy problem among a large set of math, physics, and similar articles and (3) the question leading off this section is not unlike others I have seen and reflects the same comprehensibility issues. Despite my education, I too have found many articles to be incomprehensible. I am one guy with a modest amount of time on my hands and I can only insert myself into a few articles. Perhaps by improving this one and explaining why and what policy ideas guided it, I can inspire others similarly situated to do the same.

Symplectic Integrators are really an applied physics topic or, more specifically, computational physics, rather than one considered within theoretical physics. Mathematicians also have plenty to say about symplectic spaces, but its physicists and research engineers who use those integrators. The nature of the technique draws upon theoretical concepts in the domain of physics known as analytical dynamics. In "modern times" advanced analytical dynamics has become merged with differential geometry, topology, functional analysis, and related topics and at its higher realization seems to be in the process of being absorbed by what is called geometric mechanics. Einstein's general relativity is an early example of the merger of dynamics with differential geometry, but it took a few decades before this took off as a general trend. At about the time it did take off, the higher reaches of mathematics had started to transition from proofs dominated by set theoretic concepts to to being ever more reliant upon the reconceptualization of everything in terms of differential geometry, topology and other related disciplines which have been ascendant for some time. As a result, there is nowadays an extensive reliance upon multiple advanced vocabularies and concept sets which can, if used, render the simplest ideas quite incomprehensible to most readers. It is my personal observation (I avoid calling it merely a belief) that projects like Wikipedia have been negatively impacted by over use of esoteric and excessively abstract representations of things. You should not need multiple PhDs to extract useful content from every article on physics in particular. For a very on-point, and but harsher, statement of this position addressed to a professional audience, see the nicely worded Introduction in this paper: David Hestenes, Hamiltonian Mechanics with Geometric Calculus In: Z. Oziewicz et al (eds.), Spinors, Twistors, Clifford Algebras and Quantum Deformations, Kluwer: Dordercht/Boston (1993), 203–214. The article in isolation was found at http://geocalc.clas.asu.edu/pdf/Hamilton.pdf, accessed 3/25/2013.

I am a non-specialized theoretical physicist. Despite that in graduate school my department included a mathematical physicist of some renown, I was only exposed to differential geometry in the context of general relativity and had no prior background in it prior to my late-in-life entry into a PhD program. Like pretty much any PhD graduate, I nevertheless have enough training to proceed further as needed and unravel things as necessary. (P.S. my advisor (condensed matter theory), brilliant though he was, did not support my taking modern math courses though I did sneak off an take at least one<g>). Anyway, while I doubt it will be particularly necessary to improve this article, I welcome the collaboration of anyone with a stronger background in differential geometry and/or geometric mechanics. Having spent serious time in front of a blackboard teaching physics to undergraduates, however, I am confident that I can distinguish incomprehensible explanations from comprehensible ones and that things do not always have to be dumbed down to be made comprehensible. Pedagogy is about being clear and complete and knowing when an advanced concept is really helpful versus justified only to refer the reader to sources of deeper insight.

I start soon and will go slowly. Comments on the above will be cheerfully received, but lets spend most of our time working on article content. Thanks scanyon (talk) 20:31, 24 March 2013 (UTC) (revised scanyon (talk) 21:01, 26 March 2013 (UTC))

I am glad that someone said what needed to be said. Yes, this is a plague. I too have a PhD in physics. Why do I have to go on a scavenger hunt of obscure terms like "wedge product" and "cohomology" to figure out what an article is saying? Wikipedia of all places should default to a lower level language. Why is it that every physics article is written like a graduate-level math textbook??? 74.140.199.156 (talk) 03:10, 20 May 2021 (UTC)

A Request for more practical info

"Symplectic integrators possess as a conserved quantity a Hamiltonian which is slightly perturbed from the original one. By virtue of these advantages".

I am a mere pusher of 1s and 0s, and don't really know what the advantage is of mostly conserving the original Hamiltonian. In addition to the statement above, I would love to see a more "practical" statement of the virtues. Maybe something like: "the error is therefore of the order (dt^n)" or "numeric stability is improved significantly over non-SI methods", or whatever will help me compare this usefully to say Runge-Kutta or other non-SI methods. — Preceding unsigned comment added by 76.104.205.238 (talk) 18:43, 31 August 2011 (UTC)

The application "In Plasma Physics" seems oddly specific. Especially with involvement of Yoshida's trick used to construct higher order splitting. Could Yoshida's trick be discussed in a separate section? Could there perhaps be some example of a practical implementation (pseudo-code)? Could there be some application/examples involving non-autonomous systems? Egeris (talk) 19:22, 2 September 2019 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Symplectic integrator/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Need to add: partitioned Runge-Kutta, Gauss-Legendre, history, connection with generating functions, advantages of symplectic integrators like KAM / Nekhoroshev results, approximate energy preservation, linear error growth. Problems with variable step size. Mention that there are more splitting methods. -- Jitse Niesen (talk) 13:54, 2 August 2007 (UTC)

Last edited at 13:54, 2 August 2007 (UTC). Substituted at 07:29, 30 April 2016 (UTC)

problem in "A second-order example" section?

As stated in the page, the proposed second-order example should match the Verlet method. However, by doing the math, it seems that this is only true with c1 = 1 and c2 = 0, and not the other way around (d1 and d2 are OK). I tried to use c1 = 0 and c2 = 1 (as it is currently stated in the article), but this results in integration steps that I can clearly see that do not hold for constant accelerations. Am I interpreting something wrong?

I was also confused by this However, a look at previous equations reveals that the higher order indices are applied first! Since they are operators, they act from right to left. Therefore, Verlet's method is recovered by first setting c=1, d=1/2, then again with c=0, d=1/2. Therefore, the coefficients are ok, but the "simplified equations" are, in my opinion, wrong. They should regressive equations, giving e.g. v_{i-1} in terms of x_i and v_i, to be run from i=k to i=1. --Daniel (talk) 11:29, 17 September 2021 (UTC)

This seems an important **open issue** in this page. I ran across it when implementing the fourth-order version, and running into trouble. I then saw this discussion: https://scicomp.stackexchange.com/questions/20533/test-of-3rd-order-vs-4th-order-symplectic-integrator-with-strange-result ...which agrees with your statement that the order should be reversed. How to best do this without changing too much at the same time? — Preceding unsigned comment added by 92.225.65.73 (talk) 11:41, 15 November 2023 (UTC)

Note: I now at least inserted a remark in the 'Examples' section stating that the index is always traversed in decreasing order. It would be, however, much better, if the operator statements were rephrased so they have the ordering with the lowest index value to the far right (applied first), because that would then make everything below very easy to follow. It would also require a change in the order of the listed coefficients further below. — Preceding unsigned comment added by 92.225.65.73 (talk) 11:55, 15 November 2023 (UTC)