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Entropy 2017, 19(10), 518;

Connecting Information Geometry and Geometric Mechanics

Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
Departments of Psychology and Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
Author to whom correspondence should be addressed.
Received: 13 July 2017 / Revised: 10 September 2017 / Accepted: 20 September 2017 / Published: 27 September 2017
(This article belongs to the Special Issue Information Geometry II)
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The divergence function in information geometry, and the discrete Lagrangian in discrete geometric mechanics each induce a differential geometric structure on the product manifold Q × Q . We aim to investigate the relationship between these two objects, and the fundamental role that duality, in the form of Legendre transforms, plays in both fields. By establishing an analogy between these two approaches, we will show how a fruitful cross-fertilization of techniques may arise from switching formulations based on the cotangent bundle T * Q (as in geometric mechanics) and the tangent bundle T Q (as in information geometry). In particular, we establish, through variational error analysis, that the divergence function agrees with the exact discrete Lagrangian up to third order if and only if Q is a Hessian manifold. View Full-Text
Keywords: Lagrangian; Hamiltonian; Legendre map; symplectic form; divergence function; generating function; Hessian manifold Lagrangian; Hamiltonian; Legendre map; symplectic form; divergence function; generating function; Hessian manifold
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Leok, M.; Zhang, J. Connecting Information Geometry and Geometric Mechanics. Entropy 2017, 19, 518.

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