Equation Solving with NSolve

I have the code:

G = 0.1;
β = 0.5;
ωc = 10;

integralgamma[ω_, τ_] :=
4 G ω Exp[-ω/ωc] ((1 -
Cos[ω τ])/ω^(2)) Coth[β ω/2];

mem : γ[τ_] :=
mem = NIntegrate[
integralgamma[ω, τ], {ω, 0, Infinity},
MaxRecursion -> 15, PrecisionGoal -> 2, Method -> "LocalAdaptive"]

Table[NSolve[{Sin[χ]*Cos[α] ==
1/Sqrt *Exp[-γ[τ]] &&
Sin[χ]*Sin[α] == 1/Sqrt*Exp[-γ[τ]] &&
Cos[χ] == 1/Sqrt,
0 <= χ <= π &&
0 <= α <= 2 π}, {χ, α}], {τ, 0, 2,
0.1}]

But I get the errors:

NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

{{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \ {}, {}, {}, {}}

When I remove the time dependent exponential factor, I am able to solve the equation. What's going on?

Edit:

Another try:

I first table the expression for the time-dependent exponential factor:

FunctionList[χ_, α_] =
Table[Sin[χ]*Cos[α] ==
1/Sqrt *Exp[-γ[τ]] &&
Sin[χ]*Sin[α] == 1/Sqrt*Exp[-γ[τ]] &&
Cos[χ] == 1/Sqrt, {τ, 0, 2, 0.1}];

It works; I get a list with 21 elements.

When run:

NSolve[{#,
0 <= χ <= π &&
0 <= α <= 2 π}, {χ, α}] & /@
FunctionList[χ, α],

I get the same errors. Suggestions?

• You are giving NSolve a system of 3 equations in 2 variables (treating tau as a parameter). Are you sure that these equations even have a solution? If your system is over determined, it makes sense that NSolve cannot find solutions. – Sjoerd Smit Nov 7 '16 at 16:24
• Possibly the problem is that the equations are overdetermined. They are consistent, up to numeric fuzz, but it is possible that the methods being used do not manage to catch that in the presence of approximate machine floats. – Daniel Lichtblau Nov 7 '16 at 16:24
• @SjoerdSmit This command, for instance, works: NSolve[{Sin[\[Chi]]*Cos[\[Alpha]] == 1/Sqrt && Sin[\[Chi]]*Sin[\[Alpha]] == 1/Sqrt && Cos[\[Chi]] == 1/Sqrt, 0 <= \[Chi] <= Pi && 0 <= \[Alpha] <= 2 Pi}, {\[Chi], \[Alpha]}] – Junaid Aftab Nov 7 '16 at 16:28
• Try any values on the right hand sides of the three equations in so far as the sum of their squares is one. This is just me trying to solve for the polar and azimuthual angles for points specified on a unit sphere in spherical coordinates. – Junaid Aftab Nov 7 '16 at 16:29
• If the aforementioned command works, shouldn't it work for the case I'm trying to solve: tacking on time dependent exponential factors, which effectively means I am trying to solve more than one equation. – Junaid Aftab Nov 7 '16 at 16:30