Abstract
The statistical bundle is the set of couples () of a probability density Q and a random variable W such that . On a finite state space, we assume Q to be a probability density with respect to the uniform probability and give an affine atlas of charts such that the resulting manifold is a model for Information Geometry. Velocity and acceleration of a one-dimensional statistical model are computed in this set up. The Euler–Lagrange equations are derived from the Lagrange action integral. An example Lagrangian using minus the entropy as potential energy is briefly discussed.
Highlights
The set-up of classical Lagrangian Mechanics is a finite-dimensional Riemannian manifold.For example, see the monographs by V.I
The statistical bundle is a semi-algebraic subset of R2N ; i.e., it is defined by algebraic equations and strict inequalities
Following the original construction of Amari’s Information Geometry [4], we have defined on the statistical bundle a manifold structure which is both an affine and a Riemannian manifold
Summary
The set-up of classical Lagrangian Mechanics is a finite-dimensional Riemannian manifold. Preliminary versions of this paper have been presented at the SigmaPhy2017 Conference held in Corfu, Greece, 10–14 July 2017, and at a seminar held at Collegio Carlo Alberto, Moncalieri, on 5 September 2017. In these early versions, we did not refer to Leok and Zhang’s work, which we were unaware of at that time.
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