AI 中文总结
研究由一般拉格朗日作用泛函诱导的最优传输距离，推导统一变分框架连接多种等价公式，在标准凸性假设下建立等价性，揭示瓦瑟斯坦空间的哈密顿结构，扩展经典理论并提供统一视角。
AI 中文摘要
我们研究了由一般拉格朗日作用泛函诱导的最优传输距离。扩展经典的蒙日 - 康托罗维奇和贝纳穆 - 布雷尼尔理论，我们推导了一个统一的变分框架，该框架连接了诱导传输距离的几种等价公式，包括拉格朗日、欧拉、凸优化、哈密顿 - 雅可比对偶和哈密顿流公式。在拉格朗日的标准凸性假设下，我们通过变分论证和凸对偶性建立了这些公式的等价性。所得的最优性系统揭示了瓦瑟斯坦空间上的自然哈密顿结构，为最优传输、哈密顿动力学和最优控制之间提供了直接联系。所提出的框架将经典的二次成本理论扩展到一般拉格朗日成本，并为分析由作用泛函生成的传输度量提供了统一的视角。
英文摘要
We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a unified variational framework that connects several equivalent formulations of the induced transport distance, including Lagrangian, Eulerian, convex optimization, Hamilton - Jacobi dual, and Hamiltonian flow formulations. Under standard convexity assumptions on the Lagrangian, we establish the equivalence of these formulations through variational arguments and convex duality. The resulting optimality system reveals a natural Hamiltonian structure on the Wasserstein space, providing a direct link between optimal transport, Hamiltonian dynamics, and optimal control. The proposed framework extends the classical quadratic-cost theory to general Lagrangian costs and offers a unified perspective for the analysis of transport metrics generated by action functionals.
Comments19 pages; Revised version: clarified terminology, including the distinction between distance vs cost, and refined the discussion on the Riemann metric tensor
