Jet SingleTime Lagrange Geometry and Its ApplicationsISBN: 9781118127551
216 pages
August 2011

Jet SingleTime Lagrange Geometry and Its Applications guides readers through the advantages of jet singletime 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 timedependent Lagrange geometry to the classical timedependent Lagrange geometry. A collection of jet geometrical objects are also examined such as dtensors, relativistic timedependent semisprays, harmonic curves, and nonlinear connections. Numerous applications, including the gravitational theory developed by both the BerwaldMoór 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 firstorder ordinary differential equation systems are regarded as harmonic curves on 1jet spaces. In addition, numerous opportunities are provided for readers to gain skill in applying jet singletime Lagrange geometry to solve a wide range of problems.
Extensively classroomtested to ensure an accessible presentation, Jet SingleTime 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 modelproviding geometric structures.
Part I. The Jet SingleTime Lagrange Geometry
1. Jet geometrical objects depending on a relativistic time 3
1.1 dTensors on the 1jet space J1(R, M) 4
1.2 Relativistic timedependent semisprays. Harmonic curves 6
1.3 Jet nonlinear connection. Adapted bases 11
1.4 Relativistic timedependent and jet nonlinear connections 16
2. Deflection dtensor identities in the relativistic timedependent Lagrange geometry 19
2.1 The adapted components of jet Γlinear connections 19
2.2 Local torsion and curvature dtensors 24
2.3 Local Ricci identities and nonmetrical deflection dtensors 30
3. Local Bianchi identities in the relativistic timedependent Lagrange geometry 33
3.1 The adapted components of hnormal Γlinear connections 33
3.2 Deflection dtensor identities and local Bianchi identities for dconnections of Cartan type 37
4. The jet RiemannLagrange geometry of the relativistic timedependent Lagrange spaces 43
4.1 Relativistic timedependent Lagrange spaces 44
4.2 The canonical nonlinear connection 45
4.3 The Cartan canonical metrical linear connection 48
4.4 Relativistic timedependent Lagrangian electromagnetism 50
4.5 Jet relativistic timedependent Lagrangian gravitational theory 51
5. The jet singletime electrodynamics 57
5.1 RiemannLagrange geometry on the jet singletime 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 singletime FinslerLagrange geometry for the rheonomic BerwaldMoór metric of order three 65
6.1 Preliminary notations and formulas 66
6.2 The rheonomic BerwaldMoór metric of order three 67
6.3 Cartan canonical linear connection. DTorsions and dcurvatures 69
6.4 Geometrical field theories produced by the rheonomic BerwaldMoór metric of order three 72
7. Jet local singletime FinslerLagrange approach for the rheonomic BerwaldMoór metric of order four 77
7.1 Preliminary notations and formulas 78
7.2 The rheonomic BerwaldMoór metric of order four 79
7.3 Cartan canonical linear connection. DTorsions and dcurvatures 81
7.4 Geometrical gravitational theory produced by the rheonomic BerwaldMoór metric of order four 84
7.5 Some physical remarks and comments 87
7.6 Geometric dynamics of plasma in jet spaces with rheonomic BerwaldMoór metric of order four 89
8. The jet local singletime FinslerLagrange 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. dtorsions and dcurvatures 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, dtorsions and dcurvatures 103
9.4 Geometrical field model produced by the jet conformal Minkowski metric 115
Part II. Applications of the Jet SingleTime 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 1jet spaces. Canonical nonlinear connections 123
10.3 from first order ODEs systems and Riemannian metrics to geometrical objects on 1jet spaces 127
10.4 Geometrical objects produced on 1jet spaces by first order ODEs systems and pairs of Euclidian metrics. Jet YangMills energy 129
11. Jet singletime Lagrange geometry applied to the Lorenz atmospheric ODEs system 141
11.1 Jet RiemannLagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction 135
11.2 YangMills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system 138
12. Jet singletime Lagrange geometry applied to evolution ODEs systems from Economy 141
12.1 Jet RiemannLagrange geometry for Kaldor nonlinear cyclical model in business 141
12.2 Jet RiemannLagrange geometry for TobinBenhabibMiyao economic evolution model 144
13. Some evolution equations from Theoretical Biology and their singletime Lagrange geometrization on 1jet spaces 147
13.1 Jet RiemannLagrange geometry for a cancer cell population model in biology 148
13.2 The jet RiemannLagrange geometry of the infection by human immunodeficiency virus (HIV1) evolution model 151
13.3 From calcium oscillations ODEs systems to jet YangMills energies 154
14. Jet geometrical objects produced by linear ODEs systems and higher order ODEs 169
14.1 Jet RiemannLagrange geometry produced by a nonhomogenous linear ODEs system or order one 169
14.2 Jet RiemannLagrange geometry produced by a higher order ODE 172
14.3 RiemannLagrange geometry produced by a nonhomogenous linear ODE of higher order 175
15. Jet singletime geometrical extension of the KCCinvariants 179
References 185
Index 191
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 thirtyfive journal articles on jet RiemannLagrange geometry and its applications.
"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 uptodate source of the field, but also invitations to understand and to approach some deep and difficult problems in mathematics and physics.” (Zentralblatt MATH, 2012)