Hi
As hinted by @acer, this was a problem in the simplifier, an undesired side effect of recent developments in simplifying trigonometric functions. The issue is fixed, and the fix is distributed for everybody using Maple 2024 within the Maplesoft Physics Updates v.1702 or newer. As usual, to install the Updates, open Maple and input Physics:-Version(latest)

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

Open the help page for PDEtools entering ?PDEtools: check the section on Symmetry and related solution PDE commands; you will see there SymmetryCommutator.

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

Download dsolve_Lie.mw

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

Download dsolve_Lie_methods.mw

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

No, there is not such a convert routine. It is a good idea, though. Meantime, suppose some g1 is the metric in the context of DifferentialGeometry, for example
g1 := evalDG((dx &t dx) + (dx &t dy) + (dy &t dx) + x*y*((dz &t dw) + (dw &t dz)))

The following produces the Matrix you are asking for:
Matrix(4, {op(map(u -> op(u[1]) = u[2], op([1, 2], g1)))})

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

This one is fixed, and the fix distributed for everybody using Maple 2024 within the Maplesoft Physics Updates v.1693. As usual, to install the Updates, open Maple and input Physics:-Version(latest)

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

This is fixed; quick work by Austin from Maplesoft. The fix is included in the latest Maplesoft Physics Updates for Maple 2024. As usual, to install, open Maple and input Physics:-Version(latest);

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

I am curious about this question. The operations where having a dimension set is relevant are: to sum over the repeated indices or to see the tensor's components (not a closed tensorial expression but the actual components), both of which require, of course, a dimension set. So, if you want to perform computations regardless of the dimension, it suffices to have it set AND do not perform those two operations, which anyway would have no meaning unless you set a dimension. In summary, I do not see what computation you want to do without having a dimension set that you cannot do with the dimension set.

Can you clarify that? Or even better, post a worksheet where you show what you are trying to do that you cannot do now due to having a dimension set.

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

Hi
I was out of town, seeing this today. If something is posted, related to the Maplesoft Physics Updates, and I am not responding here, I suggest you email me at physics@maplesoft.com.

The problem you mention happens only with the non-default value kernelopts('assertlevel'=2), and the non-default method=_RETURNVERBOSE. It shouldn't, of course. But it sometimes can happen. Fixes in other areas of the Maple library (and new developments in the Physics package) are the contents of the Maplesoft Physics Updates. As mentioned other times, although having fixes in places right away significantly enhances the Maple experience, the fixes themselves sometimes require further touches. Your post here is an example.

Details.
Integrating fixes of the Maple library into the Maplesoft Physics Updates package includes complete testing against the test suite at Maplesoft. Also, these fixes are the ones already installed in the Maple library being developed at Maplesoft. Still, as it happens with software, a fix may pass OK when tested and still have an issue, which, when noted, gets fixed, and another test is added to the test suite to avoid recurring into the same problem.

When you install the Maplesoft Physics Updates, all the fixes to different areas of the Maple library included are automatically active, regardless of whether you input with(Physics). For this reason, people install it, occasionally finding a new issue (as in this case of your post); others may prefer to wait till the next Maple release and not have the ongoing fixes installed right away.

I will include a resolution for this new issue in the next version of the Physics Updates to be uploaded today or tomorrow.

______________________________________________________________

UPDATE Jan/18: The issue mentioned is resolved within the Maplesoft Physics Updates v.1642 just uploaded, or any newer version of it. As usual, to install it, open Maple and input Physics:-Version(latest).

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

For a system like this one, you can solve for x as a function of y then solve the remaining equation that involves only y, or the other way around. When you pass [x, y] you are stating an ordering. Type [y, x] and you see the reverse. Try with {x, y} or not indicating the dependent variables at all, and dsolve's internals will decide the ordering for you.

Besides answering your question, Kitonum remarked that both solutions are correct; yes, that is the case. Generally speaking, you can think of the 'decoupling mechanism' as reducing a coupled system of equations into a subsystem that involves only one unknown and equations that involve the rest. Solve the system for that single unknown and remove it from the starting problem.

Generally speaking, which unknown you remove first is just a matter of choice. For linear systems, like the one you show, @nm , any choice is (generally speaking ...) as good and so to speak, equivalent. For nonlinear systems, depending on which dependent variable you eliminate first, the choice can be a good one or, on the other extreme, disastrous. For both linear and nonlinear systems, you can indicate your preferred choice passing the unknowns as a list; this is explained in dsolve,details, also in the help pages of pdsolve. And if you don't pass the unknowns as a list, dsolve (and pdsolve) will decide for you according to several considerations that I skip here.

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

The help page for Physics:-Vectors and related pages/examples are clear, though they are all related to 3D vectors. If you want to work with 4D "vectors", that is what people frequently call a tensor of one index in that you don't have the traditional vectorial operations (as those found in Physics:-Vectors in 3D) in 4 dimensions.

About using Tensors (see the help page ?Physics,Tensors) you can always set the metric to Euclidean and don't need to fool around with raising and lowering indices. It is simpler than what is implied in what you said.

Moreover, even if the metric is not Euclidean, you also don't need to use contravariant and covariant indices: input everything with covariant indices and the translation of repeated indices into one covariant and the other contravariant is done automatically by the system. So you can, but you do not need to input contravariant indices.

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

If you use the Physics Maple package, just input Setup(signature), and check ?Setup to change the signature as you may prefer (any integer dimension above 1 is fine). If your question is in general, you need to diagonalize the metric and check the sign of its trace. See details e.g. in Metric signature explanation - Physics Stackexchange.

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

From the equations you show, I see you are using the Maplesoft Physics package, not grtensor (which is not from Maplesoft).

So, input Setup(usecoordinatesastensorindices = true), and after that T1[r, r], T1[t,t] give you the rr, tt components of T1, or for the case, of any tensor you define using the Define command with a tensorial equation as the ones you show (for details about that, see Sec 2.c of the help page ?Physics,Tensors)..

Alternatively, say r is the first coordinate in the list of coordinates (you see the ordering of the coordinates entering Coordinates()), and say t is the second one; then T1[1, 1] and T1[2, 2] give you the rr and tt components too.

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

Note that conjugate(a + b + c) does not automatically distribute over the sum. If that is what you want, in the Maple context, you need to use the expand command. It's been noted (see some of the replies) that being N* a vector, expand was required twice; you asked why: it was an oversight, and applying it only once should have sufficed. That is fixed within the Maplesoft Physics Updates v.1510 or newer. So now map(expand, N*) performs the expansion to the end. As usual, to install the Updates, open a worksheet and input Physics:-Version(latest);

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

One way to do that: substitute l_1 = l_1(k[1], k[2], k[3], a[1], a[2]),  l_2= l_2(.. same dependency as l_1 ...), etc., so indicate complete dependency on all the parameters. Then add the following differential equations: diff(l_1(k[1], k[2], k[3], a[1], a[2]), k[2]) = 0, diff(l_1(k[1], k[2], k[3], a[1], a[2]), k[2]) = 0, etc., the same for l__2(...), etc. where each "derivative = 0" indicates the "not dependency of each l_j on the parameters you say they don't depend". 

Now, you have augmented the system, formulating the problem correctly; now, call PDEtools:-Solve on that augmented system.

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