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Develops the theory of jet single-time Lagrange geometry and presents modern-day applicationsJet Single-Time Lagrange Geometry and Its Applications guides readers through the advantages of jet single-time Lagrange geometry for geometrical modeling. With comprehensive chapters that outline topics ranging in complexity from basic to advanced, the book explores current and emerging applications across a broad range of fields, including mathematics, theoretical and atmospheric physics, economics, and theoretical biology.The authors begin by presenting basic theoretical concepts that serve as the foundation for understanding how and why the discussed theory works. Subusequent chapters compare the geometrical and physical aspects of jet relativistic time-dependent Lagrange geometry to the classical time-dependent Lagrange geometry. A collection of jet geometrical objects are also examined such as d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections. Numerous applications, including the gravitational theory developed by both the Berwald-Moór metric and the Chernov metric, are also presented.Throughout the book, the authors offer numerous examples that illustrate how the theory is put into practice, and they also present numerous applications in which the solutions of first-order ordinary differential equation systems are regarded as harmonic curves on 1-jet spaces. In addition, numerous opportunities are provided for readers to gain skill in applying jet single-time Lagrange geometry to solve a wide range of problems.Extensively classroom-tested to ensure an accessible presentation, Jet Single-Time Lagrange Geometry and Its Applications is an excellent book for courses on differential geometry, relativity theory, and mathematical models at the graduate level. The book also serves as an excellent reference for researchers, professionals, and academics in physics, biology, mathematics, and economics who would like to learn more about model-providing geometric structures.

Jet Single-Time Lagrange Geometry and Its Applications

Preface.Part I. The Jet Single-Time Lagrange Geometry1. Jet geometrical objects depending on a relativistic time 31.1 d-Tensors on the 1-jet space J1(R, M) 41.2 Relativistic time-dependent semisprays. Harmonic curves 61.3 Jet nonlinear connection. Adapted bases 111.4 Relativistic time-dependent and jet nonlinear connections 162. Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry 192.1 The adapted components of jet Gamma-linear connections 192.2 Local torsion and curvature d-tensors 242.3 Local Ricci identities and nonmetrical deflection d-tensors 303. Local Bianchi identities in the relativistic time-dependent Lagrange geometry 333.1 The adapted components of h-normal Gamma-linear connections 333.2 Deflection d-tensor identities and local Bianchi identities for d-connections of Cartan type 374. The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces 434.1 Relativistic time-dependent Lagrange spaces 444.2 The canonical nonlinear connection 454.3 The Cartan canonical metrical linear connection 484.4 Relativistic time-dependent Lagrangian electromagnetism 504.5 Jet relativistic time-dependent Lagrangian gravitational theory 515. The jet single-time electrodynamics 575.1 Riemann-Lagrange geometry on the jet single-time Lagrange space of electrodynamics EDLn/1 585.2 Geometrical Maxwell equations of EDLn/1 615.3 Geometrical Einstein equations on EDLn/1 626. Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moór metric of order three 656.1 Preliminary notations and formulas 666.2 The rheonomic Berwald-Moór metric of order three 676.3 Cartan canonical linear connection. D-Torsions and d-curvatures 696.4 Geometrical field theories produced by the rheonomic Berwald-Moór metric of order three 727. Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moór metric of order four 777.1 Preliminary notations and formulas 787.2 The rheonomic Berwald-Moór metric of order four 797.3 Cartan canonical linear connection. D-Torsions and d-curvatures 817.4 Geometrical gravitational theory produced by the rheonomic Berwald-Moór metric of order four 847.5 Some physical remarks and comments 877.6 Geometric dynamics of plasma in jet spaces with rheonomic Berwald-Moór metric of order four 898. The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four 998.1 Preliminary notations and formulas 1008.2 The rheonomic Chernov metric of order four 1018.3 Cartan canonical linear connection. d-torsions and d-curvatures 1038.4 Applications of the rheonomic Chernov metric of order four 1059. Jet Finslerian geometry of the conformal Minkowski metric 1099.1 Introduction 1099.2 The canonical nonlinear connection of the model 1119.3 Cartan canonical linear connection, d-torsions and d-curvatures 1039.4 Geometrical field model produced by the jet conformal Minkowski metric 115Part II. Applications of the Jet Single-Time Lagrange Geometry10. Geometrical objects produced by a nonlinear ODEs system of first order and a pair of Riemannian metrics 12110.1 Historical aspects 12110.2 Solutions of ODEs systems of order one as harmonic curves on 1-jet spaces. Canonical nonlinear connections 12310.3 from first order ODEs systems and Riemannian metrics to geometrical objects on 1-jet spaces 12710.4 Geometrical objects produced on 1-jet spaces by first order ODEs systems and pairs of Euclidian metrics. Jet Yang-Mills energy 12911. Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system 14111.1 Jet Riemann-Lagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction 13511.2 Yang-Mills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system 13812. Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy 14112.1 Jet Riemann-Lagrange geometry for Kaldor nonlinear cyclical model in business 14112.2 Jet Riemann-Lagrange geometry for Tobin-Benhabib-Miyao economic evolution model 14413. Some evolution equations from Theoretical Biology and their single-time Lagrange geometrization on 1-jet spaces 14713.1 Jet Riemann-Lagrange geometry for a cancer cell population model in biology 14813.2 The jet Riemann-Lagrange geometry of the infection by human immunodeficiency virus (HIV-1) evolution model 15113.3 From calcium oscillations ODEs systems to jet Yang-Mills energies 15414. Jet geometrical objects produced by linear ODEs systems and higher order ODEs 16914.1 Jet Riemann-Lagrange geometry produced by a non-homogenous linear ODEs system or order one 16914.2 Jet Riemann-Lagrange geometry produced by a higher order ODE 17214.3 Riemann-Lagrange geometry produced by a non-homogenous linear ODE of higher order 17515. Jet single-time geometrical extension of the KCC-invariants 179References 185Index 191