Hi @nm, good catch. It's was a subtle issue. It is fixed in the Udpates v.1723 for Maple 2024. As usual, to install, open Maple 2024 and input Physics:-Version(latest).

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

So maybe I do not understand your question.

You define R[a, b] = P, where P is a tensorial expression with a and b as free indices. OK.

Next, you compute the Taylor expansion of R[1, 1] and equate that to 0. But nowhere in your worksheet do I see why R[1,1] would be equal to 0. Anyway, that implies an ODE for f(r).

Next, you compute the Taylor expansion of R[~1, 1] and equate that to 0.  Naturally, R[1, 1] <> R[~1, 1] because R[~1, 1] = g_[~1, ~a]*R[a, 1] (including there the sum over the repeated index a) and g_[~1, ~a] <> 1. So, naturally, the ODE you get for f(r) from the Taylor expansion of R[~1, 1] is different from the one you got from the expansion of R[1, 1].

Why are you equating two different things to 0, and then expecting them to result in the same equation for f(r)? Or even why are you suggesting that R[1, 1] = R[~1, 1]?

Guessing a bit, if you want to determine the value of f(r) such that R[1, 1] = R[~1, 1], then do this:
dsolve(Gtaylor(R[~1, 1] - R[1, 1] = 0, b, 2), f(r));
                                    f(r) = -1

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

This is Maple TTY running with no initialization whatsoever, and with the latest Maplesoft Physics Updates: there is no crash. I suggest you try removing any initialization file.

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

The Physics Updates do work. What didn't work was the download mechanism. It looks fixed now in v.1714 - I just downloaded it (input Physics:-Version(latest), as usual)

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

Hi @nm
I removed all initialization files to be sure, and run it directly in TTY to be sure also that there was no GUI element interfering. Even so, I cannot reproduce your problem, and I do have, as you, the Maplesoft Physics Updates installed, as you see in this pic. The first line, anames(); shows there is no initialization file or previous assignment. I'd suggest you check that in your computer.

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

You can define differential operators and operate with them algebraically, with multiplication understood as the operator's application. You go from multiplication to application using Physics:-Library:-ApplyProductsOfDifferentialOperators. All that is explained in the help page of Physics:-Setup, the section for differential operators.

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

Hi
A fix for this is distributed for everybody using Maple 2024 within the Maplesoft Physics Updates v.1705 or newer. As usual, to install the Updates, open Maple and input Physics:-Version(latest) So now we have:

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

Yes, you can construct these operators and compute with them algebraically, check the help page ?Physics,Setup, the section on differentialoperators. You can see an illustration of how to work with these operators in the post on "Quantum Commutation Rules - Basics"

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

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