Jet Single-Time Lagrange Geometry and Its

This book describes the main geometrical and physical aspects that differentiate two geometrical theories: the presented jet relativistic time-dependent Lagrangian geometry and the classical time-dependent Lagrangian geometry. An emphasis on the jet transformation group of the first approach is more general and natural than the transformation group used in the second approach, mainly due to the fact that the last approach ignores temporal reparametrizations. In addition, the presented transformation group is appropriate for the construction of corresponding relativistic time-dependent Lagrangian geometrical field theories (gravitational and electromagnetic). The developed theory is further illustrated with numerous applications in mathematics, theoretical physics (including electrodynamics, relativity, and electromagnetism), atmospheric physics, economics, and theoretical biology. The geometrical Maxwell and Einstein equations presented in the book naturally generalize the already classical Maxwell and Einstein equations from the Miron-Anastasiei theory. The extended geometrical Einstein equations that govern the jet single-time Lagrange gravitational theory are canonical, and the electromagnetic d-tensor is produced from the metrical deflection d-tensors, all preceding entities being derived only from the given jet Lagrangian via its attached Cartan canonical Gamma-linear connection. The basic elements of the Kosambi-Cartan-Chern theory on the 1-jet space that extend the KCC tangent space approach are featured at the end of the book. Chapters are written in an introductory and gradual manner and contain numerous examples and open problems. An index of notions makes the main concepts of the theory and of the applications easy to locate.

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

Preface	p. xi
The Jet Single-Time Lagrange Geometry
Jet geometrical objects depending on a relativistic time	p. 3
d-tensors on the 1-jet space J1 (R, M)	p. 4
Relativistic time-dependent semisprays. Harmonic curves	p. 6
Jet nonlinear connections. Adapted bases	p. 11
Relativistic time-dependent semisprays and jet nonlinear connections	p. 16
Deflection d-tensor identities in the relativistic time-dependent Lagrange geometry	p. 19
The adapted components of jet P-linear connections	p. 19
Local torsion and curvature d-tensors	p. 24
Local Ricci identities and nonmedical deflection d-tensors	p. 30
Local Bianchi identities in the relativistic time-dependent Lagrange geometry	p. 33
The adapted components of h-normal $$$-linear connections	p. 33
Deflection d-tensor identities and local Bianchi identities for d-connections of Cartan type	p. 37
The jet Riemann-Lagrange geometry of the relativistic time-dependent Lagrange spaces	p. 43
Relativistic time-dependent Lagrange spaces	p. 44
The canonical nonlinear connection	p. 45
The Cartan canonical metrical linear connection	p. 48
Relativistic time-dependent Lagrangian electromagnetism	p. 50
The jet single-lime electromagnetic field	p. 50
Geometrical Maxwell equations	p. 51
Jet relativistic time-dependent Lagrangian gravitational theory	p. 52
The jet single-time gravitational field	p. 52
Geometrical Einstein equations and conservation laws	p. 53
The jet single-time electrodynamics	p. 57
Riemann-Lagrange geometry on the jet single-time Lagrange space of electrodynamics $$$	p. 58
Geometrical Maxwell equations on $$$	p. 61
Geometrical Einstein equations on $$$	p. 62
Jet local single-time Finsler-Lagrange geometry for the rheonomic Berwald-Moor metric of order three	p. 65
Preliminary notations and formulas	p. 66
The rheonomic Berwald-Moor metric of order three	p. 67
Cartan canonical linear connection, d-lorsions and d-curvatures	p. 69
Geometrical field theories produced by the rheonomic Berwald-Moor metric of order three	p. 72
Geometrical gravitational theory	p. 72
Geometrical electromagnetic theory	p. 76
Jet local single-time Finsler-Lagrange approach for the rheonomic Berwald-Moor metric of order four	p. 77
Preliminary notations and formulas	p. 78
The rheonomic Berwald-Moor metric of order four	p. 79
Cartan canonical linear connection, d-torsions and d-curvatures	p. 81
Geometrical gravitational theory produced by the rheonomic Berwald-Moor metric of order four	p. 84
Some physical remarks and comments	p. 87
On gravitational theory	p. 87
On electromagnetic theory	p. 89
Geometric dynamics of plasma in jet spaces with rheonomic Berwald-Moor metric of order four	p. 89
Introduction	p. 89
Generalized Lagrange geometrical approach of the non-isotropic plasma on 1 -jet spaces	p. 90
The non-isolropic plasma as a medium geometrized by the jet rheonomic Berwald-Moor metric of order four	p. 96
The jet local single-time Finsler-Lagrange geometry induced by the rheonomic Chernov metric of order four	p. 99
Preliminary notations and formulas	p. 100
The rheonomic Chernov metric of order four	p. 101
Cartan canonical linear connection, d-torsions and d-curvatures	p. 103
Applications of the rheonomic Chernov metric of order four	p. 105
Geometrical gravitational theory	p. 105
Geometrical electromagnetic theory	p. 108
Jet Finslerian geometry of the conformal Minkowski metric	p. 109
Introduction	p. 109
The canonical nonlinear connection of the model	p. 111
Cartan canonical linear connection, d-torsidns and d-curvatures	p. 113
Geometrical field model produced by the jet conformal Minkowski metric	p. 115
Gravitational-like geometrical model	p. 115
Related electromagnetic model considerations	p. 118
Applications of the Jet Single-Time Lagrange Geometry
Geometrical objects produced by a nonlinear ODEs system of first-order and a pair of Riemannian metrics	p. 121
Historical aspects	p. 121
Solutions of ODEs systems of order one as harmonic curves on 1-jet spaces. Canonical nonlinear connections	p. 123
From first-order ODEs systems and Riemannian metrics to geometrical objects on 1-jet spaces	p. 127
Geometrical objects produced on 1-jet spaces by first-order ODEs systems and pairs of Euclidian metrics. Jet Yang-Mills energy	p. 129
Jet single-time Lagrange geometry applied to the Lorenz atmospheric ODEs system	p. 135
Jet Riemann-Lagrange geometry produced by the Lorenz simplified model of Rossby gravity wave interaction	p. 135
Yang-Mills energetic hypersurfaces of constant level produced by the Lorenz atmospheric ODEs system	p. 138
Jet single-time Lagrange geometry applied to evolution ODEs systems from Economy	p. 141
Jet Riemann-Lagrange geometry for Kaldor nonlinear cyclical model in business	p. 141
Jet Riemann-Lagrange geometry for Tobin-Benhabib-Miyao economic evolution model	p. 144
Some evolution equations from Theoretical Biology and their single-time Lagrange gebmetrization on 1-jet spaces	p. 147
Jet Riemann-Lagrange geometry for a cancer cell population model in biology	p. 148
The jet Riemann-Lagrange geometry of the infection by human immunodeficiency virus (HIV-1) evolution model	p. 151
From calcium oscillations ODEs systems to jet Yang-Mills energies	p. 154
Intracellular calcium oscillations induced by self-modulation of the inositol 1,4,5-triphosphate signal	p. 155
Calcium oscillations in a model involving endoplasmic reticulum, mitochondria, and cytosolic proteins	p. 161
Yang-Mills energetic surfaces of constant level. Theoretical biological interpretations	p. 167
Jet geometrical objects produced by linear ODEs systems and higher-order ODEs	p. 169
Jet Riemann-Lagrange geometry produced by a non-homogenous linear ODEs system of order one	p. 169
Jet Riemann-Lagrange geometry produced by a higher-order ODE	p. 172
Riemann-Lagrange geometry produced by a non-homogenous linear ODE of higher-order	p. 175
Jet single-time geometrical extension of the KCC-invariants	p. 179
References	p. 185
Index	p. 191
Table of Contents provided by Ingram. All Rights Reserved.

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Jet Single-Time Lagrange Geometry and Its Applications

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