@dharr and @Pascal4QM,
Good catch. There was indeed one place, in one internal routine, where the old dot operator `.`, blind regarding Not(commutative) objects, was in use. This is now fixed,

The fix distributed to everyone using Maple 2023 within the Maplesoft Physics Updates v.1431 and newer.

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

@Thomas Richard 
Yes, if you include those two equations within that list, you have simplify also not introducing csch and sech.

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

@Thiago_Rangel7 
Yes, 1/cot(z) -> tan(z) looks more convenient; it is a different thing though; also working that way since ages, not new in Maple 2023. I am currently not working in simplify but will take a look, maybe it is something easy to address.

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

Hi
This post is now updated with the finished version of this material that appears in Maple 2023 as the help page Courseware-Support, Mechanics. The material - 70 solved problems on a computer algebra worksheet covering most of the syllabus of Mechanics courses - runs also in Maple 2022.2 after installing the last Maplesoft Physics Updates for Maple 2022, that is, v.1409, uploaded today. 

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

@prldb 

You asked a very similar (basically the same) question in 2020. The answer today would be the same, only emphasizing that the ordering of the NP null vectors that conform a tetrad depends on the signature, basically on where you put the timelike component in the metric. For the signature used in Maple, which is also the one used in Stephanie's book, the ordering is n, m, mb and l also in the book. And yes the convention used by Maple for the NP null vectors is the existing one (standard) - also used in Stephanie's book. All this is shown in that answer of 2020, also explained in the link I mentioned in the previous reply "What to take care of when entering a tetrad".

Now, the tetrad is defined up to an arbitrary rotation in 4 dimensions, from which you can mix everything and write a new tetrad with the imaginary unit anywhere you want. That doesn't change the fact that n, m, mb, l form a tetrad when you put them in the correct order according to the signature, and doesn't change the definitions of these NP vectors that you can see, for example with n_, via

Try TensorArray([%]) with some simplfier and you see all their defining equations are satisfied, regardless of where you put the imaginary unit.

Regarding the optimized tetrads shown in Stephanie's book: Maple is not using those tetrads - as said in a previous reply above, those book's tetrads are coded, but won't be used until they are reviewed. Instead, Maple computes a tetrad from scratch as explained in ?Tetrads:-IsTetrad, and as said there you could also enter any other different tetrad that you may prefer and the system will work with it the same way. Or if you prefer to perform transformations to optimize your tetrad in any particular way, see ?Tetrads:-TransformTetrad.

Regarding Maple's tetrad, which is constructed from the NP vectors, the standard ones, in the right ordering according to the signature used in Stephanie's book, or regarding any other tetrad, you can always test whether it is or not a correct tetrad using Tetrad:-IsTetrad,  or checking whether it satisfies the tetrad equations, basically input TensorArray(e_[definition]) and you see, by eye, if it is or not a correct tetrad, regardless of where you put complex components. Here is for the metric you are mentioning:

In summary: the tetrad computed by Maple is correct, the NP null vectors are correct too, they are the standard ones, and the conventions for how to order the NP vectors to form a tetrad are those shown in the book too, as shown step-by-step in the answer I gave to you during 2020. The only thing that is not a match to Stephanie's book is that Maple computes a tetrad that is not as the one shown there, where the case is that tetrads are defined up to an arbitrary 4D rotation.

I'm sorry but also need to move to other topics.

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

@sand15 

Physics significantly enlarges the computational domain, i.e. what can be part of the input and output. That can be felt as "a world within Maple", I suppose.

Regarding your question about the Updates, quoting from Maplesoft Physics Updates, the answer is

"... You can take advantage of this ongoing work by downloading the research version of Physics as it is updated with improvements, fixes, and the very latest new developments. ..".

This ongoing work happens on a daily basis, so you have new versions of the Maplesoft Physics Updates - on average - say 365 times per year. You see why it cannot be "in sync" with Maple's version numbers (typically only two per year).

If by 'such a special package' you refer to 'updating the package on the web around the clock', I prefer to work in close contact with people who use the package(s) and to accomplish that - in practice - requires quick feedback that people can use right away (thus, the Maplesoft Physics Updates).  As a result, Physics is what it is and the project just moves forward at its own rhythm.

Finally, note that the Maplesoft Physics Updates include not just Physics, but also Typesetting, Differential Equations, Mathematical Functions, some others, and frequently also bug fixes of things that interfere with the computational functionality in any of those areas. 

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

@mmcdara 

Your approach is also interesting in that it shows that changing the value of the summation index is - conceptually and so for the computer - a change of variables. By the way, IntegrationTools:-Change performs changes of variables calling PDEtools:-dchange.

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

@C_R 
It is unfortunate for me that you didn't bet. :)

To the side, if we disregard the word we use, and instead focus on the concept, sums and integrals have summation and integration variables and in both cases (also in sums) what we are doing is (conceptually) "a change of variables".

Incidentally, the very first, tinny routine from which dchange evolved later on into a full-change-of-variables command is this one for "changing variables in Sums", in connection with a paper I was co-authoring years ago in plasma physics.

Anyway, I can understand your comment today. I will try to remember to add an example of this in ?dchange.

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

@prldb 

Could you please also take a look at this other question: "What to take care of when entering a tetrad" - I think it is related to your question now.

Independently, yes the documentation needs to be clear about these things, I will take care of that.

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

@prldb 
I just opened one of your worksheets and clicked !!! - saw the following, which I suppose answers one of your questions. The arguments of P you are entering are not in the correct ordering:

Have a good weekend.

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

Hi
Sorry for the delay in replying. We just finished the forthcoming Maple 2023 and that took all my working cycles. I intend to review Tetrads stuff and the database of solutions to Einstein's equations starting mid-March. We do have the tetrads shown in the book coded, but that is not active (pending a revision); instead, they are computed on the fly, frequently resulting in tetrads not as optimized as the ones shown in the book. I will write here again about that, and if I find time before that will answer the specific questions you pose. Although I am not using Maple 2021 - changes there, it is an older version, are not practically feasible. The answers will concern Maple 2022, either clarifying something if that is the case or fixing if that were the case.

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

@itsme 

It is lprint(Typesetting:-Typeset(%)), not lprint(Typesetting:-Typeset(r)), not Typesetting:-Typeset(%). Yes, Typeset has an uneval restriction on its argument, but using % you workaround that. Your latex/Anticommutator works, but what you are sending to latex is its evaluation, ie. AntiCommutator, not Anticommutator - therefore the output you see. 

Now on the core of your reply: your latex/AntiCommutator does not work because there exists a print/AntiCommutator that takes precedence; probably shouldn't. if you want to make it work, then after you define your latex/AntiCommutator, do these two things unassign(print/AntiCommutator), and if you previously called latex with the same input (as you do in your worksheet), remember to latex(AntiCommutator(b__1, b__2), forget), so that the previous result is erased from latex's remember table.

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

@albivaldmaple 

The answer is further below; see the help page for Physics:-TensorArray.

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

@albivaldmaple 
That is so because the system allows you to compute with the tensors, or with their matrix form according to what you need. What is then missing in your worksheet is as in this image (worksheet attached at the end), just add a call to TensorArray and you get the expected result as (11):

Download Verify_lorentz_transformation_(reviewed).mw

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

@segfault 
If you want coordinates as arbitrary as x1, x2, x3, x4, that is what you get with Setup(coordinates = X), but that are named nu, mu, eta, xi, then input Setup(coordinates = (X = [nu, mu, eta, xi])) as explained in the help pages ?Coordinates and ?Physics,Tensor, also indicated in the first reply. 

By the way, everybody has the chance to (can) answer, regardless of others having answered.

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