Jet Single-Time Lagrange Geometry and Its Applications

VLADIMIR BALAN, PhD, is Professor in the Department of Mathematics and Informatics at the University Politehnica of Bucharest, Romania. He has published extensively in his areas of research interest, which include harmonic maps, variational problems in fiber bundles, and generalized gauge theory and its applications in mechanics and mathematical physics. MIRCEA NEAGU, PhD, is Assistant Professor in the Department of Algebra, Geometry, and Differential Equations at the Transilvania University of Bra¿¿ov, Romania. He is the author of more than thirty-five journal articles on jet Riemann-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 Γ-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  Γ-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

"It will be a happy addition to the references on this topic, and it will replace some books that now are hard to find." (Mathematical Reviews, 1 January 2013)"The book should be of interest to specialists as well as to beginners, who can find here not only an up-to-date source of the field, but also invitations to understand and to approach some deep and difficult problems in mathematics and physics." (Zentralblatt MATH, 2012)