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.

  • $\begingroup$ Without the actual equations, it's hard to say anything useful. $\endgroup$ May 16, 2020 at 11:20
  • $\begingroup$ @J.M. I thought there was a general method for doing this procedure regardless. Isn’t there? $\endgroup$
    – Y2H
    May 16, 2020 at 11:35
  • 1
    $\begingroup$ 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. $\endgroup$ May 16, 2020 at 12:01
  • $\begingroup$ @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? $\endgroup$
    – Y2H
    May 16, 2020 at 12:15

1 Answer 1


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);


(* 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)) *)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.