By Vladimir Balan
Develops the speculation of jet single-time Lagrange geometry and provides modern day applicationsJet Single-Time Lagrange Geometry and Its purposes courses readers in the course of the benefits of jet single-time Lagrange geometry for geometrical modeling. With finished chapters that define subject matters ranging in complexity from easy to complex, the publication explores present and rising functions throughout a large variety of fields, together with arithmetic, theoretical and atmospheric physics, economics, and theoretical biology.The authors start via offering uncomplicated theoretical innovations that function the basis for figuring out how and why the mentioned thought works. Subusequent chapters evaluate the geometrical and actual points of jet relativistic time-dependent Lagrange geometry to the classical time-dependent Lagrange geometry. a suite of jet geometrical items also are tested similar to d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections. a variety of functions, together with the gravitational thought built via either the Berwald-Mo?r metric and the Chernov metric, also are presented.Throughout the ebook, the authors supply quite a few examples that illustrate how the speculation is placed into perform, and so they current a number of functions within which the strategies of first-order traditional differential equation structures are considered as harmonic curves on 1-jet areas. additionally, a variety of possibilities are supplied for readers to achieve ability in using jet single-time Lagrange geometry to resolve a variety of problems.Extensively classroom-tested to make sure an obtainable presentation, Jet Single-Time Lagrange Geometry and Its functions is a superb publication for classes on differential geometry, relativity concept, and mathematical types on the graduate point. The publication additionally serves as a superb reference for researchers, execs, and teachers in physics, biology, arithmetic, and economics who want to examine extra approximately model-providing geometric buildings.
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Additional resources for Jet Single-Time Lagrange Geometry and Its Applications
Sample text
The developed generalized-theoretical field theory is governed by natural geometrical Maxwell and Einstein equations, which generalize the classical ones. In order to have a clear exposition of our ideas, we point out that, in our geometrical development, we use the following two distinct notions:' 1. time-dependent Lagrangian function = asmooth function L on J 1 ( K , M ) ; 1 We note that if L is a Lagrangian function on J1 (R, M), then C = L\Jh\\, metric on K, represents a Lagrangian on J 1 (M, M).
The torsion of a normal iV-linear connection in the Miron's case is characterized only by five effective adapted components (see [56], p. 24). Remark 57. For the Berwald Γ-linear connection BY, associated to the metrics hn(t) and φ^(χ), all adapted local torsion d-tensors vanish, except n( fc ) (l)ij _ mk ~ ^mijVl n m ' where ÍK^ÍJ {x) are the classical local curvature components of the semi-Riemannian metric φ^(χ). The expressions of the local curvature d-tensors of an arbitrary Γ-linear connection, together with the particular properties of an /i-normal Γ-linear connection, imply a considerable reduction (from fifteen to five) of the effective local curvature d-tensors that characterize an /i-normal Γ-linear connection.
1) is of the form (1)(1 "'WOT >(t,x*,rf) := i - ^ =Ä»(*) fly (i,**,rf), ■dy\dy[ where/i 11 = l / / i n > Oandg¿j(í,x fc ,yi) is a distinguished tensor on E, symmetric, of rank n = dim M, and having a constant signature, is called a Kronecker h-regular Lagrangian function with respect to the temporal Riemannian metric h = {h\\). In this context, we introduce the following concept: Definition 71. A pair RRUl = ( ^ ( Ε , Μ ) , ^ , where n = d i m M and L is a Kronecker Λ,-regular Lagrangian function, is called a relativistic time-dependent (nonautonomous or rheonomic) Lagrange space or a jet single-time Lagrange space.