Jet single-time Lagrange geometry and its applications

"This book describes the main geometrical and physical aspects that differentiate two geometrical theories: the presented jet relativistic time-dependent Lagrangian geometry and the classical time-dependent Lagrangian geometry. An emphasis on the jet transformation group of the first approach is more general and natural than the transformation group used in the second approach, mainly due to the fact that the last approach ignores temporal reparametrizations. In addition, the presented transformation group is appropriate for the construction of corresponding relativistic time-dependent Lagrangian geometrical field theories (gravitational and electromagnetic). The developed theory is further illustrated with numerous applications in mathematics, theoretical physics (including electrodynamics, relativity, and electromagnetism), atmospheric physics, economics, and theoretical biology. The geometrical Maxwell and Einstein equations presented in the book naturally generalize the already classical Maxwell and Einstein equations from the Miron-Anastasiei theory. The extended geometrical Einstein equations that govern the jet single-time Lagrange gravitational theory are canonical, and the electromagnetic d-tensor is produced from the metrical deflection d-tensors, all preceding entities being derived only from the given jet Lagrangian via its attached Cartan canonical Gamma-linear connection. The basic elements of the Kosambi-Cartan-Chern theory on the 1-jet space that extend the KCC tangent space approach are featured at the end of the book. Chapters are written in an introductory and gradual manner and contain numerous examples and open problems. An index of notions makes the main concepts of the theory and of the applications easy to locate"--

Front Matter -- The Jet Single-Time Lagrange Geometry. Jet geometrical objects depending on a relativistic time -- Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry -- Local Bianchi identities in the relativistic time-dependent Lagrange geometry -- The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces -- The jet single-time electrodynamics -- Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Mo̤r metric of order three -- Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Mo̤r metric of order four -- The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four -- Jet Finslerian geometry of the conformal Minkowski metric -- Applications of the Jet Single-Time Lagrange Geometry. Geometrical objects produced by a nonlinear ODEs system of first-order and a pair of Riemannian metrics -- Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system -- Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy -- Some evolution equations from Theoretical Biology and their single-time Lagrange geom…