Geodesics in a warped product \(B\times_fF\) of intrinsic metric spaces are examined. Since the projection of a geodesic to the base \(B\) is essentially independent of the fibre, conservative mechanics makes sense in any intrinsic metric space. Let \(B\) and \(F\) be Hadamard spaces. The main result here states that \(B\times_fF\) is a Hadamard space for any positive convex function \(f\) on \(B\). The same statement holds for the class of complete metric spaces of curvature at most \(0\) in the sense of Alexandrov.