By Radu, Dragos Hrimiuc, Kideo Shimada and Sorin V. Sabau Miron
The name of this booklet isn't a surprise for individuals operating within the box of Analytical Mechanics. notwithstanding, the geometric innovations of Lagrange area and Hamilton area are thoroughly new. The geometry of Lagrange areas, brought and studied in ,, was once ext- sively tested within the final 20 years by means of geometers and physicists from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and U.S.A. Many foreign meetings have been dedicated to debate this topic, court cases and monographs have been released , , , ,... a wide sector of applicability of this geometry is advised via the connections to Biology, Mechanics, and Physics and likewise by means of its basic environment as a generalization of Finsler and Riemannian geometries. the concept that of Hamilton house, brought in ,  used to be intensively studied in , , ,... and it's been winning, as a geometrical concept of the Ham- tonian functionality the elemental entity in Mechanics and Physics. The classical Legendre’s duality makes attainable a ordinary connection among Lagrange and - miltonspaces. It finds new innovations and geometrical gadgets of Hamilton areas which are twin to these that are related in Lagrange areas. Following this duality Cartan areas brought and studied in , ,..., are, approximately conversing, the Legendre duals of sure Finsler areas , , . The above arguments make this monograph a continuation of , , emphasizing the Hamilton geometry.
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