The Geometry of Hamilton and Lagrange Spaces (Fundamental Theories of Physics, Volume 118) (Fundamental Theories of Physics)

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. This Monograph Presents For The First Time The Foundations Of Hamilton Geometry. The Concept Of Hamilton Space, Introduced By The First Author And Investigated By The Authors, Opens A New Domain In Differential Geometry With Large Applications In Mechanics, Physics, Optimal Control, Etc. The Book Consists Of Thirteen Chapters. The First Three Chapters Present The Topics Of The Tangent Bundle Geometry, Finsler And Lagrange Spaces. Chapters 4-7 Are Devoted To The Construction Of Geometry Of Hamilton Spaces And The Duality Between These Spaces And Lagrange Spaces. The Dual Of A Finsler Space Is A Cartan Space. Even This Notion Is Completely New, Its Geometry Has The Same Symmetry And Beauty As That Of Finsler Spaces. Chapter 8 Deals With Symplectic Transformations Of Cotangent Bundle. The Last Five Chapters Present, For The First Time, The Geometrical Theory And Applications Of Higher-order Hamilton Spaces. In Particular, The Case Of Order Two Is Presented In Detail. Audience: Mathematicians, Geometers, Physicists, And Mechanicians. This Volume Can Also Be Recommended As A Supplementary Graduate Text. 1 Geometry Of Tangent Bundle 1 -- 1.1 Manifold T M 1 -- 1.2 Homogeneity 4 -- 1.3 Semisprays On The Manifold T M 7 -- 1.4 Nonlinear Connections 9 -- 1.5 Structures Ip, If 13 -- 1.6 D-tensor Algebra 18 -- 1.7 N-linear Connections 20 -- 1.8 Torsion And Curvature 23 -- 1.9 Parallelism. Structure Equations 26 -- 2 Finsler Spaces 31 -- 2.1 Finsler Metrics 31 -- 2.2 Geometric Objects Of The Space F[superscript N] 34 -- 2.3 Geodesics 38 -- 2.4 Canonical Spray. Cartan Nonlinear Connection 40 -- 2.5 Metrical Cartan Connection 42 -- 2.6 Parallelism. Structure Equations 45 -- 2.7 Remarkable Connections Of Finsler Spaces 48 -- 2.8 Special Finsler Manifolds 49 -- 2.9 Almost Kahlerian Model Of A Finsler Manifold 55 -- 3 Lagrange Spaces 63 -- 3.1 Notion Of Lagrange Space 63 -- 3.2 Variational Problem Euler-lagrange Equations 65 -- 3.3 Canonical Semispray. Nonlinear Connection 67 -- 3.4 Hamilton-jacobi Equations 70 -- 3.5 Structures Ip And If Of The Lagrange Space L[superscript N] 71 -- 3.6 Almost Kahlerian Model Of The Space L[superscript N] 73 -- 3.7 Metrical N-linear Connections 75 -- 3.8 Gravitational And Electromagnetic Fields 80 -- 3.9 Lagrange Space Of Electrodynamics 83 -- 3.10 Generalized Lagrange Spaces 84 -- 4 Geometry Of Cotangent Bundle 87 -- 4.1 Bundle (t* M, [pi]*, M) 87 -- 4.2 Poisson Brackets. The Hamiltonian Systems 89 -- 4.3 Homogeneity 93 -- 4.4 Nonlinear Connections 96 -- 4.5 Distinguished Vector And Covector Fields 99 -- 4.6 Almost Product Structure Ip. The Metrical Structure G. The Almost Complex Structure If 101 -- 4.7 D-tensor Algebra. N-linear Connections 103 -- 4.8 Torsion And Curvature 106 -- 4.9 Coefficients Of An N-linear Connection 107 -- 4.10 Local Expressions Of D-tensors Of Torsion And Curvature 110 -- 4.11 Parallelism. Horizontal And Vertical Paths 112 -- 4.12 Structure Equations Of An N-linear Connection. Bianchi Identities 116 -- 5 Hamilton Spaces 119 -- 5.1 Spaces G H[superscript N] 119 -- 5.2 N-metrical Connections In G H[superscript N] 121 -- 5.3 N-lift Of G H[superscript N] 123 -- 5.4 Hamilton Spaces 124 -- 5.5 Canonical Nonlinear Connection Of The Space H[superscript N] 127 -- 5.6 Canonical Metrical Connection Of Hamilton Space H[superscript N] 128 -- 5.7 Structure Equations Of Ct (n). Bianchi Identities 130 -- 5.8 Parallelism. Horizontal And Vertical Paths 131 -- 5.9 Hamilton Spaces Of Electrodynamics 133 -- 5.10 Almost Kahlerian Model Of An Hamilton Space 136 -- 6 Cartan Spaces 139 -- 6.1 Notion Of Cartan Space 139 -- 6.2 Properties Of The Fundamental Function K Of Cartan Space C[superscript N] 142 -- 6.3 Canonical Nonlinear Connection Of A Cartan Space 143 -- 6.4 Canonical Metrical Connection 144 -- 6.5 Structure Equations. Bianchi Identities 148 -- 6.6 Special N-linear Connections Of Cartan Space C[superscript N] 150 -- 6.7 Some Special Cartan Spaces 152 -- 6.8 Parallelism In Cartan Space. Horizontal And Vertical Paths 154 -- 6.9 Almost Kahlerian Model Of A Cartan Space 156 -- 7 Duality Between Lagrange And Hamilton Spaces 159 -- 7.1 Lagrange-hamilton L-duality 159 -- 7.2 L-dual Nonlinear Connections 163 -- 7.3 L-dual D-connections 168 -- 7.4 Finsler-cartan L-duality 173 -- 7.5 Berwald Connection For Cartan Spaces. Landsberg And Berwald Spaces. Locally Minkowski Spaces 179 -- 7.6 Applications Of The L-duality 184 -- 8 Symplectic Transformations Of The Differential Geometry Of T* M 189 -- 8.1 Connection-pairs On Cotangent Bundle 189 -- 8.2 Special Linear Connections On T* M 195 -- 8.3 Homogeneous Case 201 -- 8.4 F -related Connection-pairs 204 -- 8.5 F-related [phi]-connections 210 -- 8.6 Geometry Of A Homogeneous Contact Transformation 212 -- 9 Dual Bundle Of A K-osculator Bundle 219 -- 9.1 (t*[superscript K] M, [pi]*[superscript K], M) Bundle 220 -- 9.2 Dual Of The 2-osculator Bundle 227 -- 9.3 Dual Semisprays On T*[superscript 2] M 231 -- 9.4 Homogeneity 234 -- 9.5 Nonlinear Connections 237 -- 9.6 Distinguished Vector And Covector Fields 239 -- 9.7 Lie Brackets. Exterior Differentials 242 -- 9.8 Almost Product Structure Ip. The Almost Contact Structure If 244 -- 9.9 Riemannian Structures On T*[superscript 2] M 246 -- 10 Linear Connections On The Manifold T*[superscript 2] M 249 -- 10.1 D-tensor Algebra 249 -- 10.2 N-linear Connections 250 -- 10.3 Torsion And Curvature 253 -- 10.4 Coefficients Of An N-linear Connection 255 -- 10.5 H-, W[subscript 1]-, W[subscript 2]-covariant Derivatives In Local Adapted Basis 256 -- 10.6 Ricci Identities. The Local Expressions Of Curvature And Torsion 259 -- 10.7 Parallelism Of The Vector Fields On The Manifold T*[superscript 2] M 263 -- 10.8 Structure Equations Of An N-linear Connection 267 -- 11 Generalized Hamilton Spaces Of Order 2 271 -- 11.1 Spaces Gh[superscript (2)n] 271 -- 11.2 Metrical Connections In Gh[superscript (2)n]-spaces 274 -- 11.3 Lift Of A Gh-metric 277 -- 11.4 Examples Of Spaces Gh[superscript (2)n] 280 -- 12 Hamilton Spaces Of Order 2 283 -- 12.1 Spaces H[superscript (2)n] 283 -- 12.2 Canonical Presymplectic Structures And Canonical Poisson Structures 286 -- 12.3 Lagrange Spaces Of Order Two 290 -- 12.4 Variational Problem In The Spaces L[superscript (2)n] 293 -- 12.5 Legendre Mapping Determined By A Space L[superscript (2)n] 296 -- 12.6 Legendre Mapping Determined By H[superscript (2)n] 299 -- 12.7 Canonical Nonlinear Connection Of The Space H[superscript (2)n] 301 -- 12.8 Canonical Metrical N Connection Of Space H[superscript (2)n] 302 -- 12.9 Hamilton Spaces H[superscript (2)n] Of Electrodynamics 304 -- 13 Cartan Spaces Of Order 2 307 -- 13.1 C[superscript (2)n]-spaces 307 -- 13.2 Canonical Presymplectic Structure Of Space C[superscript (2)n] 309 -- 13.3 Canonical Nonlinear Connection Of C[superscript (2)n] 312 -- 13.4 Canonical Metrical Connection Of Space C[superscript (2)n] 314 -- 13.5 Parallelism Of Vector Fields. Structure Equations Of Ct (n) 317 -- 13.6 Riemannian Almost Contact Structure Of A Space C[superscript (2)n] 319. By Radu Miron ... [et Al.]. Includes Bibliographical References (p. 323-335) And Index.