Solution to v - discrepencies between sympy and mathematica

Question

I wrote a program:

import sympy as sp
# Define symbols
l, alpha, x, gamma, r, theta, beta, v = sp.symbols('l alpha x gamma r theta beta v')
c = 2.99792458 * 10**8
# Define the equation
lhs = l * sp.sin(beta)
rhs = sp.sqrt((l * alpha + x * gamma - r * theta) * sp.sqrt(1 - v**2 / c**2)) * \
 sp.sqrt((l * alpha - x * gamma + r * theta) / sp.sqrt(1 - v**2 / c**2)) / alpha
equation = sp.Eq(lhs, rhs)
# Solve the equation for v
solution = sp.solve(equation, v)
# Print the solution
print("Solutions for v:")
for sol in solution:
 print(sol)

It does not print any solutions to ( v ), whereas Mathematica does indicate that there are solutions to ( v ):

v -> (\[Sqrt](-8.98755*10^16 l^2 \[Alpha]^2 + 8.98755*10^16 x^2 \[Gamma]^2 - 1.79751*10^17 r x \[Gamma] \[Theta] + 8.98755*10^16 r^2 \[Theta]^2 + 
8.98755*10^16 l^2 \[Alpha]^2 Sin[\[Beta]]^2))/(Sqrt[-1. l^2 \[Alpha]^2 + x^2 \[Gamma]^2 - 2.r x \[Gamma] \[Theta] + r^2 \[Theta]^2 + l^2 \[Alpha]^2 Sin[\[Beta]]^2])

Should Mathematica be corrected, or should we update the SymPy library to account for this? The program runs well, but it does not output the same solution that Mathematica does.

• Davide_sd: Questions: 1. Did you write your equation correctly? Because if I manually simplify the RHS of your equation, v goes away. 2. What assumptions are there on symbols?
• Parker Emmerson: Yes, the equation is written correctly. The mathematica input is best writen as: Solve[l Sin[[Beta]] == ( Sqrt[(l [Alpha] + x [Gamma] - r [Theta]) Sqrt[ 1 - v^2/(2.99792458*10^8)^2]] Sqrt[(l \[Alpha] - x \[Gamma] + r \[Theta])/Sqrt[ 1 - v^2/(2.99792458*10^8)^2]])/[Alpha], v], which you can copy/paste directly into mathematica. Take the Reduce in Mathematica, i.e. Reduce[l Sin[[Beta]] == ( Sqrt[(l [Alpha] + x [Gamma] - r [Theta]) Sqrt[1 - v^2/(c)^2]] Sqrt[(l [Alpha] - x [Gamma] + r [Theta])/Sqrt[ 1 - v^2/(c)^2]])/[Alpha], v] for the real analysis
• Parker Emmerson: You are correct about manually canceling. We demonstrate this by writing: Sqrt[-q^2 + 2 q s - s^2 + l^2 [Alpha]^2]/[Alpha] == ( Sqrt[-(q - s - l [Alpha]) ] Sqrt[(q - s + l [Alpha])])/[Alpha] However, for some reason, Mathematica outputs a solution to the equation for v, and the most suspicious part about it is that it only works when using the speed of light in scientific notation.
• Parker Emmerson: Mathematica is closed source and has not responded to inquiries about this matter in a professional manner, thus there is no peer review for this solution. That's why I was trying to use Sympy to find another way of verifying the solution, but to no avail.
• Parker Emmerson: As far as interpreting meaning for the symbols, Latin letters, x, l, and r represent distances of arc lengths and Greek letters represent angles of those arc lengths. v represents velocity in the Lorentz coefficient, which in this equation ought cancel out with itself, and the solution to v given by mathematica is therefore a kind of algebraic-geometric velocity embedded as a hidden dimension. You can see the proof that it fits formally with conceptions of instantaneous and average velocity here: zenodo.org/records/7911884 and here: zenodo.org/records/7272772