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 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