Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S31 / 2 / a / Solution

Let the step be $h$. Forward Euler method gives $x_1=x_0+h{v}_0$, $v_1=v_0+hF(x_0)$. Applying the same map with $-h$ gives $x_2=x_1-h{v}_1=x_0-h^{2}F(x_0)$, which generally differs from $x_0$. It is therefore not its own inverse under time reversal.

A kick–drift–kick leapfrog integrator instead hasThe step $-h$ first recovers $v_{1/2}$, then $x_0$, then $v_0$, so $\boxed{{\Phi }_{-h}={\Phi }_h^{-1}}$. This assumes a position-dependent force and the same fixed step. Newtonian mechanics is reversible; respecting that structure helps avoid artificial directional bias and often improves long-term qualitative behaviour. The usual leapfrog also has a structure preserving symplectic geometry, which supports bounded oscillatory energy error, but reversibility by itself does not imply exact conservation of energy. Adaptive stepping requires special care to preserve these properties.