 Solution

The forcing is perpendicular to velocity, soStarting at unit speed therefore keeps the motion in $SM$. The horizontal part is the geodesic transport $X$, while the acceleration rotates the unit velocity at angular speed $f$. Thus the magnetic flow generator is $F=X+(f\circ \pi )V$.

For the canonical coframe of a surface unit tangent bundle, set $\mu =\alpha \wedge \theta \wedge \omega$. Its interior product of a differential form with the two relevant fields isThe structure equations giveThe Cartan formula for the Lie derivative consequently gives ${\mathcal{L}}_X\mu ={\mathcal{L}}_V\mu =0$. For the variable vertical coefficient, use the full product rule:Indeed $f$ is pulled back from the base, so $Vf=0$ and $df$ has only $\alpha ,\theta$ components. HenceThe Liouville volume of a surface geodesic flow is therefore preserved by this magnetic modification for every smooth $f$, without requiring the Anosov property.