Abstract: For surfaces of revolution Clairaut's theorem gives a first integral for geodesics: $r \cos\theta =$ constant, where $r$ is the distance from the axis to the profile curve and $\theta$ is the angle the geodesic makes with the latitude circles. We have generalized this to warped products $W = B\times_fF$ of metric spaces: along any geodesic $\gamma$ in $W$, $f^2v = b$ is constant, where $v$ is the speed of the projection of $\gamma$ to $F$. When $B, F$ are Riemannian manifolds, the geodesic equations have a known form: $$ \gamma_B'' = c(1/f^3) {\rm grad} f, \qquad (f^2v)' =0,$$ where $\gamma_B$ is the projection to $B$. This has the interpretation that $\gamma_B$ is a trajectory of the potential function $U = c/2f^2$. The fact that the speed of $\gamma$ is a constant $a = \sqrt{b/c}$ becomes the law of conservation of energy $u^2 + 2U = (b/c)^2$, where $u$ is the speed of $\gamma_B$. Hence, for more general metric spaces $B$, Clairaut's theorem makes it reasonable to interpret the projections of geodesics from a warped product $B\times_fF$ to $B$ as the trajectories of the potential function $U = 1/2f^2 :B \to {\bf R}$. Since we also have shown that these trajectories are independent of the choice of $F$, we can simply take $F$ to be the line or a circle. Joint work with Stephanie Alexander