@EugeneKalentev 

The Mapleprimes site manager answered me saying that the worksheet and reply "Easier using Physics", that you are asking, is lost in unrecoverable way. Fortunately, however, I've found an old TimeMachine backup of my older laptop actually containing this worksheet! I am amazed at how well this simple Macintosh TimeMachine backup system works. So I restored the worksheet and the answer, including a link to the worksheet, all in "Easier using Phyiscs".

 Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

Download Commutator_and_d.mw

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@mpre 

I know it may sound as advertisement but, if you could trust me ... it is not that. What it is: Physics in Maple 2015 is already a big step forward with regards to Physics in Maple 18, and it has been this way at every release since Maple 15. So if this package is of relevance for your work in any particular way, I strongly recommend you to update your copy of Maple.

Not less important: since Maple 18 (August of 2013), Maplesoft releases an update of the Physics package on the web every week (sometimes every day of a week). What this means:

• you do not need to wait one year (for a new release) to see a bug fix or feature request in place
• you can participate direclty in the development of the package by presenting your concrete real-work problem and when possible the package gets its functionality extended to be of use in your real case scenario.
• you have access to the latest developments in the package 1 release advanced.

And that happens for a whole year. You realize the disadvantage of staying with Maple 13. Besides, it is very difficult to help you with regards to the use of Maple 13 - the package is just not the same.

If you have the opportunity to update your copy, please feel free to post what you describe as your more complicated problem, and from there we move ahead, I may be able to help you, or adjust the package as required.

Best

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Michael_Watson 

I recently answered in Mapleprimes a question you made about a simliar issue: when using labels, the output was comming as 0 = 0 (wow with the coincidence!). As said in that reply, there is a bug in the interface (GUI). There is a workaround to this issue. In order to have the workaround installed, you need to update the Physics library that comes with the Maple 2015 installation, as usual by downloading it from the Maplesoft R&D Physics webpage. Could you please do that and let me know whether this problem with using GUI labels gets "resolved"? Thanks.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

When executing your worksheet, I do not get 0=0 in equation (19) as you show. Could you please post a separate worksheet with the output you expect from algsubs ((9), (18))? That will help understanding what is the problem you are experiencing, and from there we move ahead.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@patient 

Would you mind posting a worksheet with the input equations? That would help avoiding mistakes while reading the image you posted to retype the system. Thanks.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@alljoyland 

Give a look at the documentation (help pages for g_ and Setup): you can set the metric to what is appropriate for your problem. For example, if you set it to 'Euclidean', all the powerful tools you see working in curved spacetimes (general relativity) become available for tackling problems for instance in elasticity (what you are asking).

Independent of that, you can work with space indices, independently of working with spacetime indices. Recalling, the space part of a Minkowski metric is actually Euclidean  (Minkowski is the default type of spacetime that is set automatically when you load Physics, i.e. a flat non-curved space, not for general relativity). This use of space indices is illustrated in the page ?Physics,examples.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace 

When indexing tensors, say in a 4D spacetime, 1,2,3,4 refer to the positions of the components of the tensor in the corresponding tensor array. For example, if the tensor has 2 indices, the array is a matrix, so - say - the component [2, 3] is the component you see in the 2nd row, third column. Yes there is of course a relatioinship between these positions and the order you indcate for the coordinates as a list, so 1=t, 2=r, 3=theta, 4=phi, this is clear, for instance, in the spacetime metric (see the reviewed worksheet). And yes: the number itself means covariant, so 1 means 1st position, covariant, and ~1 means contravariant 1st position, so g_[~1, ~1] is the 1st line, 1st column, of the all-contravariant g_, and Christoffel[~1, 2, 2] means 1st index contravariant, position 1 in the array, and 2nd and 3rd index both covariant, position 2 in the array.

Try 

> Christoffel[~1, alpha, beta, array];

or 

> Christoffel[~alpha, 2, 2, array];

Instead of 'array' you can also use the keyword matrix. All these things and some more are explained in the documentation.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

This GUI issue has been tracked - and a solution circumvecting the problem is available - for that you need to update your Physics library from the usual place, the Maplesoft R&D Physics webpage.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Codge 

Wouldn't your reply suggest that the ics you are using are not convenient for producing sections with p1 > 0?

Independent of that, have you explored the option 'scene'? Note that you can use it as in 'scene = [p1 = 0..3, q2 = -2..2, p2 = -2..2]' or the like, ie you can specify the ranges for each of the three variables, together with the option '3' to produce a 3D section that you could then rotate with the mouse to have different 2D projection views

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft.

@Codge 

After writing Hamilton's equations (this Hamilton_eqs command) you could still try to solve numerically; then use DEtools[odeplot] to plot the solution for p1 and you will see, precisely, where it changes sign (for any particular set of ics you are using). Suppose now you have identified, visually, looking at the plots for p1 that you construct with odeplot - say three - chunks of time where p1 > 0. Do this: create the poincare sections for each of the chunks separatedly with three caslls to DEtools[poincare], and assign the result to some variable, say as in ps1 := poincare(.... t = chunk_1), ...), ps2 := poincare(..., t = chunk_2, ...), and so on, then produce a single plot of the three of them superposed using plots:-display([ps1, ps2, ...], options_you_may_prefer_here). Let me know please if this does not resolve your problem, where is that things gets stuck, and we move from there.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft.

@MathyMaple 

I actually saw your post on double summation, thought about redefinesum, checked things. Your problem is solved with the latest update of Physics and its redefinesum approach to these premature evaluation issues. As usual, the Physics update is available at the Maplesoft Physics R&D webpage.

I will post as well reply directly in your double summation post.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Markiyan Hirnyk 

Just for curiosity: are you a physicist or working in condensed matter physics? Giving a look at the papers you selected randomly, I think the answer to your "More concrete question" is actually found in the answer to your first question; the mathematical methods used in these papers appear to me to be a subset of the methods listed in the previous reply. 

@J F Ogilvie

Note please that the contents of this post is contextualized by the title, "Physics, one year of developments". This post is not about how powerful is the Maple intsolve command for solving integral equations (BTW I think it is powerful, as indicated by its help page), or about how relevant would be the Heun special functions.

I also realize that you feel that this Maple's integral-equation solver could be more powerful, and that you think the Heun functions are useless, and that you want to express these thoughts (BTW you have done this in various Physics post unrelated to special functions). But please allow us to dissent. There are physicists working with these functions, both in General Relativity and Quantum Mechanics - publications of 2013 and 2014. I am saying this not as a way to say "someone is right or wrong", but just to ask you for some space for a different opinion. 

Some things were you seem to be missing information. You say:

"The help files on Jacobi functions seem to provide no indication to the Jacobi elliptic functions cn(..,..), dn(..,..), sn(..,..)"

Check please the help page for JacobiPQ, or JacobiCN, ore any of the 12 ones. The functions are there and the documentation is too, there are not 11 but 12 + the JacobiAM (amplitud), and the help page is invoked in the right way,  enter ?JacobiPQ, or ?JacobiCN, etc.

You say:

"Asking help on Heun functions returns a sequence of "unknown" and "unable"

Check the help page ?Heun and the pages therein. Here again I do not find what you say (I mean the facts).

You say:

... these somewhat obscure [Heun] functions

Please note that many special function experts (Marichev, Brychov, etc;, google for this and you will get the idea) differ from you. The Heun functions are modern special functions being studied and having their properties unveiled by a growing number of expert people in the special function area.

... spheroidal functions and Lame functions

Although I would like to have more time to implement some of these spheroidal functions, I note that they are not currently used as it was the case 100+ years ago when special functions were not so developed in the mathematical language (I am not talking about Computer Algebra). As a measurement of their relevance in Physics, for instance, I recall that they are not mentioned in the 9 volumes of the Landau and Lifshitz "Course on Theoretical Physics ", nor in the main texbooks in Mathematical Methods for Physicists; to mention but three: Courant & Hilbert, "Methods of Mathematical Physics (2 vols)", Smirnov, "A Course of Higher Mathematics, (5 vols)",  Arfken & Weber, "Mathematical Methods for Physicists".

Anyway the above is just my opinion, one in a universe of opinions. I understood yours. It is all OK with the differences. I am not a believer in written debates. I also feel as said that this post on Physics: one year of developments is not quite the right framework for these topics you mentioned.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft
Editor, Computer Physics Communications

@H-R 

Have you checked the link indicated, "Relating 3D vectors and tensors"? I continue thinking that "... this answer contains all the ingredients I could imagine behind your question, related Maple input/output illustrating, and also pointers to specific free online literature related to your question."

Regarding this other issue to the side, your quote now of wiki, please do not missunderstand me but "curl(curl(A))" is in my opinion unclear tensor notation. Compare for instance with the right-hand side of the last equation shown in your image of wikipedia. Anyway let's not deviate: give a look please to "Relating 3D vectors and tensors", the Maple worksheet with input output available in that link, as well as the free-online "A Premier on Tensor Calculus" linked there too, pages 30 to 40. I understand there you will find the answer with details to you question.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Markiyan Hirnyk 

Condensed Matter Physics (CMP) is an area too vast. Quoting wikipedia, "The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists identify themselves as condensed matter physicists". 

Anyway, generally speaking, working in theoretical CMP requires the mathematical methods of Quantum Mechanics, Electrodynamics, Statistical Mechanics, and Field Theory, as well as powerful differential equation tools to handle from ODEs to PDE integrable systems. Most of these mathematical methods, but for perhaps functional integration and more advanced numerical PDE solving, are already implemented in the Maple system, and the implementation is improving at every release. This post is all about that.

Moving a bit away from theoretical CPM, Maple also has optimization, numerical integration and probability distribution algorithms that allow for Monte-Carlo simulations (including the MapleSim[Analysis][MonteCarlo] itself), quite useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, solids, etc. These tools are a bit scattered though, instead of within a package with specific tools for Monte-Carlo simulations in Physics. Similar to that, we do not have, for instance, a separate package for Statistical Physics, but then we do have the Maple Statistics package covering most of the related topics. 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft