Computational efforts in optimal transport traditionally revolvearound the squared-Euclidean cost. In this work, we choose toinvestigate the optimal transport problem between probability measureswhen the underlying metric space is non-Euclidean, or when the costfunction is understood to satisfy a least action principle,also known as a Lagrangian cost. These two generalizations are useful when connecting observations from a physical system, where the transport dynamics are influencedby the geometry of the system, such as obstacles, and allowspractitioners to incorporate a priori knowledge of theunderlying system. Examples include barriers for transport, orenforcing a certain geometry, i.e., paths must be circular.We demonstrate the effectiveness of this formulation on existingsynthetic examples in the literature, where we solve the optimaltransport problems in the absence of regularization, which is novel inthe literature. Our contributions are of computational interest, where we demonstrate the ability to efficiently compute geodesics and amortize spline-based paths. We demonstrate the effectiveness of this formulation on existing synthetic examples in the literature, where we solve the optimal transport problems in the absence of regularization.