Jet Single-Time Lagrange Geometry And Its Applications

Jet 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.

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.

Part I. The Jet Single-Time Lagrange Geometry.

1. Jet geometrical objects depending on a relativistic time.

1.1 d-Tensors on the 1-jet space J1(R, M).

1.2 Relativistic time-dependent semispra0ys. Harmonic curves.

1.3 Jet nonlinear connection. Adapted bases.

1.4 Relativistic time-dependent and jet nonlinear connections.

2. Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry.

2.1 The adapted components of jet G-linear connections.

2.2 Local torsion and curvature d-tensors.

2.3 Local Ricci identities and nonmetrical deflection d-tensors.

3. Local Bianchi identities in the relativistic time-dependent Lagrange geometry.

3.1 The adapted components of h-normal G-linear connections.

3.2 Deflection d-tensor identities and local Bianchi identities for d-connections of Cartan type,

4. The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces.

4.1 Relativistic time-dependent Lagrange spaces.

4.2 The canonical nonlinear connection.

4.3 The Cartan canonical metrical linear connection.

4.4 Relativistic time-dependent Lagrangian electromagnetism.

4.5 Jet relativistic time-dependent Lagrangian gravitational theory.

5. The jet single-time electrodynamics.

5.1 Riemann-Lagrange geometry on the jet single-time Lagrange space of electrodynamics ?DL₁.

5.2 Geometrical Maxwell equations of ?DL₁.

5.3 Geometrical Einstein equations on ?DL₁.

6. Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moór metric of order three.

6.1 Preliminary notations and formulas.

6.2 The rheonomic Berwald-Moór metric of order three.

6.3 Cartan canonical linear connection. D-Torsions and d-curvatures.

6.4 Geometrical field theories produced by the rheonomic Berwald-Moór metric of order three.

7. Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moór metric of order four.

7.1 Preliminary notations and formulas.

7.2 The rheonomic Berwald-Moór metric of order four.

7.3 Cartan canonical linear connection. D-Torsions and d-curvatures.

7.4 Geometrical gravitational theory produced by the rheonomic Berwald-Moór metric of order four.

7.5 Some physical remarks and comments.

7.6 Geometric dynamics of plasma in jet spaces with rheonomic Berwald-Moór metric of order four.

8. The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four.

8.1 Preliminary notations and formulas.

8.2 The rheonomic Chernov metric of order four.

8.3 Cartan canonical linear connection. D-Torsions and d-curvatures.

8.4 Applications of the rheonomic Chernov metric of order four.

9. Jet Finslerian geometry of the conformal Minkowski metric.

9.1 Introduction.

9.2 The canonical nonlinear connection of the model.

9.3 Cartan canonical linear connection, d-torsions and d-curvatures.

9.4 Geometrical field model produced by the jet conformal Minkowski metric.

Part II. Applications of the Jet Single-Time Lagrange Geometry.

10. Geometrical objects produced by a nonlinear ODEs system of first order and a pair of Riemannian metrics.

10.1 Historical aspects.

10.2 Solutions of ODEs systems of order one as harmonic curves on 1-jet spaces. Canonical nonlinear connections.

10.3 from first order ODEs systems and Riemannian metrics to geometrical objects on 1-jet spaces.

10.4 Geometrical objects produced on 1-jet spaces by first order ODEs systems and pairs of Euclidian metrics. Jet Yang-Mills energy.

11. Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system.

11.1 Jet Riemann-Lagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction.

11.2 Yang-Mills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system.

12. Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy.

12.1 Jet Riemann-Lagrange geometry for Kaldor nonlinear cyclical model in business.

12.2 Jet Riemann-Lagrange geometry for Tobin-Benhabib-Miyao economic evolution model.

13. Some evolution equations from Theoretical Biology and their single-time Lagrange geometrization on 1-jet spaces.

13.1 Jet Riemann-Lagrange geometry for a cancer cell population model in biology.

13.2 The jet Riemann-Lagrange geometry of the infection by human immunodeficiency virus (HIV-1) evolution model.

13.3 From calcium oscillations ODEs systems to jet Yang-Mills energies.

14. Jet geometrical objects produced by linear ODEs systems and higher order ODEs.

14.1 Jet Riemann-Lagrange geometry produced by a non-homogenous linear ODEs system or order one.

14.2 Jet Riemann-Lagrange geometry produced by a higher order ODE.

14.3 Riemann-Lagrange geometry produced by a non-homogenous linear ODE of higher order.

15. Jet single-time geometrical extension of the KCC-invariants.

References.

Index.