Liouville metric geodesic integral
= Liouville metric geodesic integral
{c}
{title2=$K=(U+V)(V\dot u^2-U\dot v^2)$}
For a surface metric $(U(u)+V(v))(du^2+dv^2)$ with $U+V>0$, an affinely parametrized <geodesic> has constant energy $E=(U+V)(\dot u^2+\dot v^2)/2$. Put $p=(U+V)\dot u$. The <Euler-Lagrange equations> give $\dot p=EU'/(U+V)$. Thus $K=p^2-2EU$ has derivative zero, proving the displayed additional quadratic integral. This separation structure is a geometric example of <Liouville integrability>.