# tracing – How to use the list of rules to solve to draw complex solutions?

I am trying to find the frequency of three circular particles connected in a circle with different spring constants and different masses. After deriving the equations of motion, I get three complex equations for w that I convert into a matrix. By setting the determinant to 0, you should be able to find w (the frequency). k, l, m, M are constants and w is a function of ka.

To simplify, I changed the exponential function into a trigonometric function. I assumed he would get some real solutions, but Mathica only found complex solutions. Then I wonder if the solutions are wrong or if I was wrong somewhere. The plot is completely empty.

Here is my code so far:

``````
In(299):= k = 9;
l = 12;
m = 2;
M = 4 ;
mat = {{m*w^2 - 2*k, k, k*Exp(-3 I*ka)}, {k, M*w^2 - (l + k),
l}, {-k*Exp(-3 I*ka), l, M*w^2 - (k - l)}};
mydet = ExpToTrig(Det(mat))
sol = Solve(mydet == 0, w)

Out(304)= 3483 + 558 w^2 - 432 w^4 + 32 w^6 - 1701 Cos(6 ka) +
324 w^2 Cos(6 ka) + 1701 I Sin(6 ka) - 324 I w^2 Sin(6 ka)

Out(305)= {{w -> -(Sqrt)(9/
2 + (1386 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (324 2^(1/3) Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) +
Sqrt(4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
1/(96 2^(
1/3)))((-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3)) + (324 I 2^(1/3) Sin(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3))}, {w -> (Sqrt)(9/2 + (
1386 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
324 2^(1/3)
Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3)/(
96 2^(1/3)) + (
324 I 2^(1/3)
Sin(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3))}, {w -> -(Sqrt)(9/
2 - (693 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (693 I 2^(1/3) Sqrt(3))/(-4478976 + 6718464 Cos(6 ka) +
Sqrt(4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (162 2^(1/3) Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) +
Sqrt(4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (162 I 2^(1/3) Sqrt(3) Cos(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
1/(
192 2^(1/3)))((-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3)) - (
1/(64 2^(1/3) Sqrt(3)))
I (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (162 I 2^(1/3) Sin(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (162 2^(1/3) Sqrt(3) Sin(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3))}, {w -> (Sqrt)(9/2 - (
693 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
693 I 2^(1/3) Sqrt(
3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
162 2^(1/3)
Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
162 I 2^(1/3) Sqrt(3)
Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3)/(
192 2^(1/3)) - (1/(64 2^(1/3) Sqrt(3)))
I (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
162 I 2^(1/3)
Sin(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
162 2^(1/3) Sqrt(3)
Sin(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3))}, {w -> -(Sqrt)(9/
2 - (693 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (693 I 2^(1/3) Sqrt(3))/(-4478976 + 6718464 Cos(6 ka) +
Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (162 2^(1/3) Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) +
Sqrt(4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (162 I 2^(1/3) Sqrt(3) Cos(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
1/(192 2^(
1/3)))((-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3)) + (
1/(64 2^(1/3) Sqrt(3)))
I (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (162 I 2^(1/3) Sin(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) + (162 2^(1/3) Sqrt(3) Sin(6 ka))/(-4478976 +
6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3))}, {w -> (Sqrt)(9/2 - (
693 2^(1/3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
693 I 2^(1/3) Sqrt(
3))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
162 2^(1/3)
Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
162 I 2^(1/3) Sqrt(3)
Cos(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(
1/3) - (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3)/(
192 2^(1/3)) + (1/(64 2^(1/3) Sqrt(3)))
I (-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) - (
162 I 2^(1/3)
Sin(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3) + (
162 2^(1/3) Sqrt(3)
Sin(6 ka))/(-4478976 + 6718464 Cos(6 ka) + Sqrt(
4 (-133056 + 31104 Cos(6 ka) -
31104 I Sin(6 ka))^3 + (-4478976 + 6718464 Cos(6 ka) -
6718464 I Sin(6 ka))^2) - 6718464 I Sin(6 ka))^(1/3))}}

ComplexListPlot(w /. sol, PlotLegends -> "Expressions")
``````

The plot is empty even though I have 6 complex solutions. I also tried Plot (w /. Sol, {ka, 0, pi}) which also gives an empty diagram. I don't get any errors with these codes, so I guess there is a problem in the way the solution is formatted.