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Article

Connecting Information Geometry and Geometric Mechanics

by 1 and 2,*
1
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
2
Departments of Psychology and Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(10), 518; https://doi.org/10.3390/e19100518
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)
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
MDPI and ACS Style

Leok, M.; Zhang, J. Connecting Information Geometry and Geometric Mechanics. Entropy 2017, 19, 518. https://doi.org/10.3390/e19100518

AMA Style

Leok M, Zhang J. Connecting Information Geometry and Geometric Mechanics. Entropy. 2017; 19(10):518. https://doi.org/10.3390/e19100518

Chicago/Turabian Style

Leok, Melvin, and Jun Zhang. 2017. "Connecting Information Geometry and Geometric Mechanics" Entropy 19, no. 10: 518. https://doi.org/10.3390/e19100518

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