The Geometry of Hamilton and Lagrange Spaces

Radu Miron, Dragos Hrimiuc, Hideo Shimada, Sorin V. Sabau (auth.)0792369262, 9780792369264, 9780306471353

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.

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Table of contents :
The geometry of tangent bundle….Pages 1-30
Finsler spaces….Pages 31-61
Lagrange spaces….Pages 63-86
The geometry of cotangent bundle….Pages 87-118
Hamilton spaces….Pages 119-137
Cartan spaces….Pages 139-158
The duality between Lagrange and Hamilton spaces….Pages 159-187
Symplectic transformations of the differential geometry of T*M ….Pages 189-218
The dual bundle of a k -osculator bundle….Pages 219-248
Linear connections on the manifold T* 2 M ….Pages 249-269
Generalized Hamilton spaces of order 2….Pages 271-282
Hamilton spaces of order 2….Pages 283-306
Cartan spaces of order 2….Pages 307-321