The title of this book is no surprise for people working in the
field of Analytical Mechanics. However, the geometric concepts of
Lagrange space and Hamilton space are completely new. The geometry
of Lagrange spaces, introduced and studied in [76],[96], was ext-
sively examined in the last two decades by geometers and physicists
from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and
U.S.A. Many international conferences were devoted to debate this
subject, proceedings and monographs were published [10], [18],
[112], [113],... A large area of applicability of this geometry is
suggested by the connections to Biology, Mechanics, and Physics and
also by its general setting as a generalization of Finsler and
Riemannian geometries. The concept of Hamilton space, introduced in
[105], [101] was intensively studied in [63], [66], [97],... and it
has been successful, as a geometric theory of the Ham- tonian
function the fundamental entity in Mechanics and Physics. The
classical Legendre's duality makes possible a natural connection
between Lagrange and - miltonspaces. It reveals new concepts and
geometrical objects of Hamilton spaces that are dual to those which
are similar in Lagrange spaces. Following this duality Cartan
spaces introduced and studied in [98], [99],..., are, roughly
speaking, the Legendre duals of certain Finsler spaces [98], [66],
[67]. The above arguments make this monograph a continuation of
[106], [113], emphasizing the Hamilton geometry.
General
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