I am learning PDE and can someone help me solve the following practice problem? I want to convert the following to a 4 x 4 first-order system $u'=f(t,u), u(0)=u_0$ and $u= [x, x', y, y']= [u_1, u_2, u_3, u_4]$.
$x''= x+2y'-b(x+a)/((x+a)^2+y^2)^{3/2} -a(x-b)/((x-b)^2+y^2)^{3/2}$ $y''= y-2x'-by/((x+a)^2+y^2)^{3/2} -ay/((x-b)^2+y^2)^{3/2}$
where $a= 0.012277471$ and $b=1-a$.
The initial conditions $x(0)= 0.994; x'(0)=0; y(0)=0; y'(0)= -2.00158510637908$ are satisfied. In this problem, $(x(t), y(t))$ is the position of a satellite orbiting around the earth and moon. There is a periodic orbit with period $T= 17.06521656015796$.
And also choose a rectangular region in the 4-dimension u space which includes the initial point $u_0$, and find a Lipschitz constant $L$ for $f$ on that region.
Thanks!