# a symplectic version of the henon-heiles system
f(x,y)=-x-2*x*y
g(x,y)=-y-x^2+y^2

par h=.05
xh=x+.5*h*xp
yh=y+.5*h*yp
xp1=xp+h*f(xh,yh)
yp1=yp+h*g(xh,yh)
x'=xh+.5*h*xp1
y'=yh+.5*h*yp1
xp'=xp1
yp'=yp1
init x=0,y=-.1475,xp=.3101,yp=0
aux h=.5*(xp*xp+yp*yp)+.5*(x^2+y^2+x^2*y-y^3/3)
@ meth=discrete,total=2000
done

The leapfrog integrator:

r(n+1/2) = r(n) + .5 h v(n)
v(n+1) = v(n) + h F(r(n+1/2))
r(n+1) = r(n) + .5 h v(n+1)