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. even though, the geometric suggestions of Lagrange house and Hamilton area are thoroughly new. The geometry of Lagrange areas, brought and studied in [76],[96], used to be ext- sively tested within the final twenty years through 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 sector of applicability of this geometry is advised via the connections to Biology, Mechanics, and Physics and likewise via its normal surroundings as a generalization of Finsler and Riemannian geometries. the idea that of Hamilton area, brought in [105], [101] was once intensively studied in [63], [66], [97],... and it's been profitable, as a geometrical concept of the Ham- tonian functionality the elemental entity in Mechanics and Physics. The classical Legendre’s duality makes attainable a usual connection among Lagrange and - miltonspaces. It unearths new ideas and geometrical items of Hamilton areas which are twin to these that are related in Lagrange areas. Following this duality Cartan areas brought and studied in [98], [99],..., are, approximately talking, the Legendre duals of sure Finsler areas [98], [66], [67]. The above arguments make this monograph a continuation of [106], [113], emphasizing the Hamilton geometry.

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**Example text**

2. 3) on TM, we obtain Consequently, the tensorial equations Wjk = 0, h ing. 3. 12), the Lie brackets U— ,7-r give an horizontal vector fields \_dx3 ox"J if and only if &ik = 0. The previous property allows to say that Rljk is the curvature tensor field of the nonlinear connection N. 14) is the torsion of the nonlinear connection N. 3. have The almost complex structure W is integrable if and only if we # V = 0, fjk = 0. 16) Proof. , we deduce dxl oyJ 5 M d \ _ , t = Rl 6 { d —~ + t- \ dyi dy J oyl Now it follows that Afw = 0 «=* {R}^ = fjk = 0}.

1) T{X, Y) = DXY - DYX - [X, Y], X, Y £ X{TM). [(XH, YH) + T(XH, Yv) + T(XV, YH) + TT(XV, Yv). 1. 2) hTT(XH, Yv) = -D^XH - [XH, Yv}», vTT(XH,Yv) = D\YV - [XH,YV]V, vT{Xv,Yv) = DVXYV - D\XV - [XV,YV]V, X,Y € X{TM). 1. The following properties hold: a) vT{XH,YH) = 0 <=*• HTM is an integrable distribution. b) h1T{XH, Yv) = 0<=^ vD»Yv = [XH, Yv]v, X, Y e X{TM), We shall say that hT(XH, YH) is /i(M)-torsion of D, thatvT(XH, YH) is torsion, etc. 2. 3) pi fi _ ft =dN[_Li ffi gi of the J (~*
*

*The previous property allows to say that Rljk is the curvature tensor field of the nonlinear connection N. 14) is the torsion of the nonlinear connection N. 3. have The almost complex structure W is integrable if and only if we # V = 0, fjk = 0. 16) Proof. , we deduce dxl oyJ 5 M d \ _ , t = Rl 6 { d —~ + t- \ dyi dy J oyl Now it follows that Afw = 0 «=* {R}^ = fjk = 0}. d. 6 d-tensor Algebra Let N be a nonlinear connection on the manifold E = TM. 2)'. 1) where XH belongs to the horizontal distribution HTM. *