@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

@maleclerc88 

The problem is resolved, you need to update your Physics library from the usual place, the Maplesoft R&D Physics webpage.

The issue: because A is a quantum opetator, A[1] operates on the first quantum number of the basis of Kets labeled A, while A[2] operats on the second quantum number of that same basis. Therefore, A[1] and A[2] commute. At the same time, the AntiCommutator rule set made A[1] and A[2] anticommute, resulting in a logical inconsistency resulting in (the absurd) A[1] A[3] = 0.

It is curious how this went undected when writing the first reply. The situation is of course unusual. Anyway the code now checks for that before concluding about commutation/anticommutation of operators, so that if A[1], A[2] anticommute, because you set a rule specifying so, then they do not commute regardless of operating on different quantum numbers of the same basis of Kets.

Then simplify(A[1] A[3]) does not return 0 (wrong) while simplify(A[1] A[3] + A[3] A[1]) does return 0 (correct). You need to update your Physics library containing this fix as said.

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

@nm 

Order is an environment variable that is assigned a number. Therefore it cannot be an optional argument: dsolve(blabla, Order = 20) would result in (arrive at dsolve as) dsolve(blabla, 6 = 20). For this reason, in dsolve,series, pdsolve,series, DifferentialAlgebra[PowerSeriesSolution], DEtools[rtaylor] - these from the top of my head but I believe that also everywhere else in the Maple system where you compute series solutions - the optional argument is called order = n, not Order = n. Just to say that Maple made its mind about this time ago, and it is consistent.

On the other hand, it is true, order = n is documented in all these help pages (pdsolve,series, DifferentialAlgebra[PowerSeriesSolution], DEtools[rtaylor], ..) but not in dsolve,series, confusing you. This is an oversight. I will make sure the option is mentioned also in dsolve,series.

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

@tomleslie 

In the help page for Physics,Vectors, I see

which shows the ordering: for spherical coordinates, phi in third place. The convention for the ranges is rather standard and found in basically all physics and vector analysis textbooks. I added an explicit sentence "where theta ranges from 0 to Pi and phi from 0 to 2*Pi right after the image in the ?Physics,Vectors help page.

VectorCalculus is not part of the Physics package. The conventions for the coordinates in the VectorCalculus package are presented in ?VectorCalculus,Coordinates and today I am not working, just perusing here, but from the top of my head (see ?VectorCalculus,SetCoordinates) you can set the ordering as you prefer, for instance via SetCoordinates(spherical[r,theta,phi]), or [r,phi,theta].

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

This new command, introduced in the library as Physics:-Library:-Assume, has more relevance than what seemed at first, so I moved it one level up, it is now one of the main commands of the Physics package. I will see if it can be made  available also as an option (e.g. redefinevariables = false) of the old assume command.

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

@Markiyan Hirnyk 

pdsolve(pde, build) returns the built form of the solution. Because a solution through separation of variables is a particular solution, it is important to make this information (the kind of separation of variables) visible. As explained in ?pdsolve, the solution is returned automatically 'built' when it is a general solution, otherwise when you receive a PDESolStruc as in this case, use the option built.. In brief: either this solution you posted, or the explicit and correct analytic-exact solution it returns when you use the option built are useful. Relevant as well, in the case of Shrodinger's equation, actually, the solutions of interest as mainly by separation of variables. 

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

@Michael_Watson 

In the reviewed worksheet I posted with the previous reply, I do not see difficulties in eqn3 - what is the problem you are experiencing? Perhaps if you share a worksheet showing the issue?

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

@acer 

The edited version you posted looks excellent Acer!

Best

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

@VicenteAtal 

I know some bits on the internals of the Maple worksheet representation but this is beyond my knowledge - I forwarded your worksheet to the Maple GUI gurus.

Best

Edgardo

Hi

Perhaps you could post a worksheet showing, say, where you are with your problem and where is that you don't know how to proceed further?

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

@J F Ogilvie 

Yes, any of these two you mentioned would serve the purpose you have, but you need to reverse the order in the equations; i.e use 

BTW, the "powerhouses" for developments like this one are the packages, that require more and more functionality.

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