Jet single-time Lagrange geometry and its applications (Q2883160)

The new book by Balan and Neagu is an excellent further development of the classical topic [\textit{R. Miron} and \textit{M. Anastasiei}, The geometry of Lagrange spaces: theory and applications. Fundamental Theories of Physics. 59. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0831.53001)]. The authors have spent many years studying and researching this field and are well qualified to produce a new monograph with the above title. The book presents the jet single-time Lagrange geometry for geometric modeling. The relativistic geometrical approach concerning geometrical and physical meaning and handling is elegant and flexible in a multitude of applications. Numerous examples are given which illustrate how the theory is put into practice. The mathematical level is high, for beginners a skilled tutor is helpful. I recommend the book for all people engaged in the hard job of mathematical modeling of complex time dependent problems in economics, biology, atmospheric physics and astrophysics for example. The main chapter headlines show the variety of the content of the book.NEWLINENEWLINEPart I. The jet single-time Lagrange geometry. 1. Jet geometrical objects depending on a relativistic time. 2. Deflection \(d\)-tensor identities in the relativistic time-dependent Lagrange geometry. 3. Local Bianchi identities in the relativistic time-dependent Lagrange geometry. 4. The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces. 5. The jet single-time electrodynamics. 6. Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moor metric of order three. 7. Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moor metric of order four. 8. The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four. 9. Jet Finslerian geometry of the conformal Minkowski metric.NEWLINENEWLINEPart II. Applications of the jet single-time Lagrange geometry.NEWLINENEWLINE10. Geometrical objects produced by a nonlinear ODEs (ordinary differential equations) system of first order and a pair of Riemannian metrics. 11. Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system. 12. Jet single-time Lagrange geometry applied to evolution ODEs systems from economy. 13. Some evolution equations from theoretical biology and their single-time Lagrange geometrization on 1-jet spaces. 14. Jet geometrical objects produced by linear ODEs systems and higher order ODEs.