Jet Single-Time Lagrange Geometry and Its Applications

Develops the theory of jet single-time Lagrange geometry and
presents modern-day 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.

Auteur

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 Brä ov, Romania. He is the
author of more than thirty-five journal articles on jet
Riemann-Lagrange geometry and its applications.

Résumé
Develops the theory of jet single-time Lagrange geometry and presents modern-day 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.

Contenu
Preface. Part I. The Jet Single-Time Lagrange Geometry

1. Jet geometrical objects depending on a relativistic time 3

1.1 d-Tensors on the 1-jet space J1(R, M) 4

1.2 Relativistic time-dependent semisprays. Harmonic curves 6

1.3 Jet nonlinear connection. Adapted bases 11

1.4 Relativistic time-dependent and jet nonlinear connections 16

2. Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry 19

2.1 The adapted components of jet -linear connections 19

2.2 Local torsion and curvature d-tensors 24

2.3 Local Ricci identities and nonmetrical deflection d-tensors 30

3. Local Bianchi identities in the relativistic time-dependent Lagrange geometry 33

3.1 The adapted components of h-normal -linear connections 33

3.2 Deflection d-tensor identities and local Bianchi identities for d-connections of Cartan type 37

4. The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces 43

4.1 Relativistic time-dependent Lagrange spaces 44

4.2 The canonical nonlinear connection 45

4.3 The Cartan canonical metrical linear connection 48

4.4 Relativistic time-dependent Lagrangian electromagnetism 50

4.5 Jet relativistic time-dependent Lagrangian gravitational theory 51

5. The jet single-time electrodynamics 57

5.1 Riemann-Lagrange geometry on the jet single-time Lagrange space of electrodynamics EDLn/1 58

5.2 Geometrical Maxwell equations of EDLn/1 61

5.3 Geometrical Einstein equations on EDLn/1 62

6. Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moór metric of order three 65

6.1 Preliminary notations and formulas 66

6.2 The rheonomic Berwald-Moór metric of order three 67

6.3 Cartan canonical linear connection. D-Torsions and d-curvatures 69

6.4 Geometrical field theories produced by the rheonomic Berwald-Moór metric of order three 72

7. Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moór metric of order four 77

7.1 Preliminary notations and formulas 78

7.2 The rheonomic Berwald-Moór metric of order four 79

7.3 Cartan canonical linear connection. D-Torsions and d-curvatures 81

7.4 Geometrical gravitational theory produced by the rheonomic Berwald-Moór metric of order four 84

7.5 Some physical remarks and comments 87

7.6 Geometric dynamics of plasma in jet spaces with rheonomic Berwald-Moór metric of order four 89

8. The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four 99

8.1 Preliminary notations and formulas 100

8.2 The rheonomic Chernov metric of order four 101

8.3 Cartan canonical linear connection. d-torsions and d-curvatures 103

8.4 Applications of the rheonomic Chernov metric of order four 105

9. Jet Finslerian geometry of the conformal Minkowski metric 109

9.1 Introduction 109

9.2 The canonical nonlinear connection of the model 111

9.3 Cartan canonical linear connection, d-torsions and d-curvatures 103

9.…