Jet Single-Time Lagrange Geometry and Its Applications

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.

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 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

9.4 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