# Jet Single-Time Lagrange Geometry and Its Applications

# Jet Single-Time Lagrange Geometry and Its Applications

ISBN: 978-1-118-14375-9

Sep 2011

224 pages

## Description

**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.

**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 Γ-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 Γ-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 EDL_{n/1} 58

5.2 Geometrical Maxwell equations of EDL_{n/1} 61

5.3 Geometrical Einstein equations on EDL_{n/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

9.4 Geometrical field model produced by the jet conformal Minkowski metric 115

**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 121**

10.1 Historical aspects 121

10.2 Solutions of ODEs systems of order one as harmonic curves on 1-jet spaces. Canonical nonlinear connections 123

10.3 from first order ODEs systems and Riemannian metrics to geometrical objects on 1-jet spaces 127

10.4 Geometrical objects produced on 1-jet spaces by first order ODEs systems and pairs of Euclidian metrics. Jet Yang-Mills energy 129

**11. Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system 141**

11.1 Jet Riemann-Lagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction 135

11.2 Yang-Mills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system 138

**12. Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy 141**

12.1 Jet Riemann-Lagrange geometry for Kaldor nonlinear cyclical model in business 141

12.2 Jet Riemann-Lagrange geometry for Tobin-Benhabib-Miyao economic evolution model 144

**13. Some evolution equations from Theoretical Biology and their single-time Lagrange geometrization on 1-jet spaces 147**

13.1 Jet Riemann-Lagrange geometry for a cancer cell population model in biology 148

13.2 The jet Riemann-Lagrange geometry of the infection by human immunodeficiency virus (HIV-1) evolution model 151

13.3 From calcium oscillations ODEs systems to jet Yang-Mills energies 154

**14. Jet geometrical objects produced by linear ODEs systems and higher order ODEs 169**

14.1 Jet Riemann-Lagrange geometry produced by a non-homogenous linear ODEs system or order one 169

14.2 Jet Riemann-Lagrange geometry produced by a higher order ODE 172

14.3 Riemann-Lagrange geometry produced by a non-homogenous linear ODE of higher order 175

**15. Jet single-time geometrical extension of the KCC-invariants 179**

References 185

Index 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)