# Re-Expressing Quantity Using Different Set of Variables

I have a quantity ds2 which is expressed in term of a set of 5 variables (let's call them t, r, θ, φ, ψ). I have 5 equations (let’s label them eqi where i=1, 2..5) relating those variables to another set of variables (let’s those T, y, Θ, Φ, Ψ). How can I express my original quantity solely using the second set?

I have tried the following pieces of code:

1. Solve
Solve[{ds2==bla bla bla, eq1 && eq2 && eq3 && eq4 && eq5}, ds2, {T, y, Θ, Φ, Ψ}, Reals]
-> "Solve: This system cannot be solved with the methods available to Solve."

1. Eliminate
Eliminate[{ds2= bla bla bla, eq1, eq2,eq3, eq4, eq5}, {T, y, Θ, Φ, Ψ}]
-> "Eliminate: Inverse functions are being used by Eliminate, so some solutions may not be found;
use Reduce for complete solution information."


Edit: Below are the explicit quantites/equations

eq1 = Xi[a] y^2 Sin[Θ]^2 == (r^2 + a^2) Sin[θ]^2;
eq2 = Φ == φ + a t/l^2;
eq3 = T == t;
eq4 = Xi[b] y^2 Cos[Θ]^2 == (r^2 + b^2) Cos[θ]^2;
eq5 = Ψ == ψ + b t/l^2;

ds2 = Sin[Θ]^2;


The actual ds is much more complicated, but even this simple case does not work and give the aforementioned error messages.

• Without the actual equations, it's hard to say anything useful. May 16 '20 at 11:20
• @J.M. I thought there was a general method for doing this procedure regardless. Isn’t there?
– Y2H
May 16 '20 at 11:35
• If the set of equations are not algebraic (like yours), there isn't. Also, D[] is reserved for the derivative, so don't use it here. May 16 '20 at 12:01
• @J.M. first of, thank you very much so far. I have updates ds2 with a much simpler case however. Could you let me know what I could do in this case if you know?
– Y2H
May 16 '20 at 12:15

Clear["Global*"]

eq1 = Xi[a] y^2 Sin[Θ]^2 == (r^2 + a^2) Sin[θ]^2;
eq2 = Φ == φ + a t/l^2;
eq3 = T == t;
eq4 = Xi[b] y^2 Cos[Θ]^2 == (r^2 + b^2) Cos[θ]^2;
eq5 = Ψ == ψ + b t/l^2;


Without some form of constraints there are many alternatives for {t, r, θ, φ, ψ}in terms of {T, y, Θ, Φ, Ψ}

sol = (Solve[
eq1 && eq2 && eq3 && eq4 && eq5,
{t, r, θ, φ, ψ}] // Simplify);

Length@sol

(* 16 *)


Further, the solutions for θ are periodic

Cases[sol, (var_ -> c_ConditionalExpression) :> {var, c[[-1]]},
Infinity] // Union

(* {{θ, C[1] ∈ Integers}} *)


Since your ds2 is initially supposed to be in terms of {t, r, θ, φ, ψ} I changed its definition. Looking at the first solution and with C[1] == 0

ds2 = Sin[θ]^2 /. sol[[1]] /. C[1] -> 0 // Simplify

(* (1/(2 (a^2 - b^2)))(a^2 - b^2 + y^2 Sin[Θ]^2 Xi[a] +
y^2 Cos[Θ]^2 Xi[
b] - √(4 (a^2 - b^2) y^2 Cos[Θ]^2 Xi[b] + (-a^2 +
b^2 + y^2 Sin[Θ]^2 Xi[a] +
y^2 Cos[Θ]^2 Xi[b])^2)) *)
`