(I am not a mathematician; I am having physics background.)
How to solve a single nonlinear algebraic equation in two variables, $x$ and $y$?
(I know that - if there are two variables, one needs two different equations to solve them.)
Equation is this :
$\frac{1}{2} (x^2+y^2) + \frac{2(1-m)}{r_1} + \frac{2m}{r_2} = C$
where $m$ and $C$ are constants and $x,y$ are two variables and
where
$r_1 = \sqrt {(x-x_1)^2+y^2}$ and
$r_2 = \sqrt {(x-x_2)^2+y^2}$
and where $x_1$ and $x_2$ are constants - i.e. two fixed values of variable $x$.
Is it possible to solve this equation analytically; if not, how to solve it numerically?
Basically my intention is to find all those $(x,y)$ points in $x-y$ plane such that LHS of this equation becomes RHS i.e. constant $C$; and that is what is the solution is.
IMPORTANT : This equation is from Forest Ray Moulton's a classic and a great book on celestial mechanics, which was published in 1914. That time there were no computers. Still author has drawn the contours (i.e. solution of this equation) in his book ! Its really amazing. My second intention behind this post is that how Moulton did this ?
