By Radu, Dragos Hrimiuc, Kideo Shimada and Sorin V. Sabau Miron

The identify of this ebook is not surprising for individuals operating within the box of Analytical Mechanics. notwithstanding, the geometric suggestions of Lagrange house and Hamilton area are thoroughly new. The geometry of Lagrange areas, brought and studied in [76],[96], was once ext- sively tested within the final 20 years by way of geometers and physicists from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and U.S.A. Many overseas meetings have been dedicated to debate this topic, lawsuits and monographs have been released [10], [18], [112], [113],... a wide quarter of applicability of this geometry is advised via the connections to Biology, Mechanics, and Physics and likewise through its normal surroundings as a generalization of Finsler and Riemannian geometries. the concept that of Hamilton area, brought in [105], [101] was once intensively studied in [63], [66], [97],... and it's been profitable, as a geometrical idea of the Ham- tonian functionality the basic entity in Mechanics and Physics. The classical Legendre’s duality makes attainable a ordinary connection among Lagrange and - miltonspaces. It finds new suggestions and geometrical gadgets of Hamilton areas which are twin to these that are comparable in Lagrange areas. Following this duality Cartan areas brought and studied in [98], [99],..., are, approximately talking, the Legendre duals of convinced Finsler areas [98], [66], [67]. The above arguments make this monograph a continuation of [106], [113], emphasizing the Hamilton geometry.

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2. 7) is just the classical law of transformation of the local coefficients of a tensor field on the base manifold M. 6)). 3. 13) are d-tensor fields of type (1,2). 7 J/'-T-T dy1 is a d-vector field. N—linear connections Let N be an a priori given nonlinear connection on the manifold E = TM. 1. e. a Kozul connection or covariant derivative) on the manifold E = TM is called an N–linear connection if: 1° D preserves by parallelism the horizontal distribution N. 2° The tangent structure J is absolute parallel with respect to D, that is DJ = 0.

For any function / € F{E). 9) D l , = y% d*,- = y% are called the h- and v-deflection tensor fields of the N-linear connection D. 3. 9)' D'j = ysUsj - tfj, tfj = 5} + y*Clsj. 9)'. The d-tensor of deflections are important in the geometry of tangent bundle. 1. The geometry of tangent bundle 23 A N-linear connection D is called of Cartan type if its tensor of deflection have the property: Dl3 = 0, dtj = S'j. 10) = 0. We will see that the canonical metrical connection in a Finsler space is of Cartan type.

1) £(c) =/>(*,§),«. ,: Remark. Starting from the property that L(x,y) = F2(x,y) is energy function of the regular Lagrangian F2, we will prove in the next chapter the following two properties. Theorem A. Along with the integral curves of the Euler-Lagrange Q equations ffr' Et (F2) = 0, y* = — we have: at dt Theorem B. 2. 1. 1) can be expressed in the form where Gl is given by (4-4) 2 Proof. 3). 4) give the integral curve of the spray The vector field S is called canonical spray of the space. 2.