using python to solve a nonlinear equation

Question

I have never used python but Mathematica can't handle the equation I am trying to solve. I am trying to solve for the variable "a" of the following equations where s, c, mu, and delta t are known parameters.

I tried doing NSolve, Solve, etc in Mathematica but it has been running for an hour with no luck. Since I am not familiar with Python, is there a way I can use Python to solve this equation for a?

Answer 1

You're not going to find an analytic solution to these equations because they're transcendental, containing a both inside and outside of a trigonometric function.

I think the trouble you're having with numerical solutions is that the range of acceptable values for a is constrained by the arcsin. Since arcsin is only defined for arguments between -1 and 1 (assuming you want a to be real), your formulas for alpha and beta require that a > s/2 and a > (s-c)/2.

In Python, you can find a zero of your third equation (rewritten in the form f(a) = 0) using the brentq function:

import numpy as np
from scipy.optimize import brentq

s = 10014.6
c = 6339.06
mu = 398600.0
dt = 780.0
def f(a):
    alpha = 2*np.arcsin(np.sqrt(s/(2*a)))
    beta = 2*np.arcsin(np.sqrt((s-c)/(2*a)))
    return alpha - beta - (np.sin(alpha)-np.sin(beta)) - np.sqrt(mu/a**3)*dt

a0 = max(s/2, (s-c)/2)
a = brentq(f, a0, 10*a0)

Edit:

To clarify the way brentq(f,a,b) works is that it searches for a zero of f on an interval [a,b]. Here, we know that a is at least max(s/2, (s-c)/2). I just guessed that 10 times that was a plausible upper bound, and that worked for the given parameters. More generally, you need to make sure that f changes sign between a and b. You can read more about how the function works in the SciPy docs.