The nonlinear oscillator $y'' + f(y)=0$ is equivalent to the
Simple harmonic motion:
$y'= -z $,
$z'= f(y)$
the modified Symplectic Euler equation are
$$y'=-z+\frac {1}{2} hf(y)$$
$$y'=f(y)+\frac {1}{2} hf_y z$$
and deduce that the coresponding approximate solution lie on the family of curves
$$2F(y)-hf(y)y+z^2=constant$$
where $F_y= f(y)$.
ans =>
for the solution of the system lie on the family of curves, i was thinking
$$\frac{d}{dt}[2F(y)-hf(y)y+z^2]= y \frac{dy}{dt} + z \frac{dz}{dt}$$
$=y(-z+\frac{1}{2} hf(y)) +z(f(y)- \frac{1}{2} h f_y z)$
but I can not do anything after that to get my answer constant.
can any genius people please help me