Information-Geometric Models in Data Analysis and Physics II

This paper continues the development of information-geometric models for data analysis and physics by focusing on their formulation and interpretation through variational principles. Building on the geometric framework introduced previously, we investigate how fundamental variational structures—such as information-theoretic functionals—naturally encode the laws of nature. In the first manuscript, we showed that a wide class of physical problems can be expressed as constrained variational problems on spaces of probability distributions, leading to geodesic flows, gradient dynamics, and generalized Hamiltonian formulations on statistical manifolds. In this second part, we extend the variational formalism by utilizing an extended metric, clarifying the geometric origin of the dynamical equations commonly used in modern physics and providing a coherent interpretation of physical laws in terms of information optimization. By emphasizing variational foundations, this paper strengthens the conceptual and mathematical links between information geometry, data analysis, and physics, and it provides a flexible framework for extending geometric methods to complex, high-dimensional, and dynamical systems.