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

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