By Vladimir Balan

Develops the speculation of jet single-time Lagrange geometry and provides modern day applicationsJet Single-Time Lagrange Geometry and Its functions courses readers during the merits of jet single-time Lagrange geometry for geometrical modeling. With complete chapters that define issues ranging in complexity from easy to complicated, the publication explores present and rising purposes throughout a huge variety of fields, together with arithmetic, theoretical and atmospheric physics, economics, and theoretical biology.The authors start through proposing easy theoretical ideas that function the root for realizing how and why the mentioned concept works. Subusequent chapters examine the geometrical and actual elements of jet relativistic time-dependent Lagrange geometry to the classical time-dependent Lagrange geometry. a suite of jet geometrical items also are tested corresponding to d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections. various functions, together with the gravitational concept built via either the Berwald-Mo?r metric and the Chernov metric, also are presented.Throughout the e-book, the authors supply quite a few examples that illustrate how the speculation is placed into perform, and so they current a variety of functions within which the strategies of first-order usual differential equation structures are considered as harmonic curves on 1-jet areas. moreover, quite a few possibilities are supplied for readers to realize ability in utilising jet single-time Lagrange geometry to unravel a variety of problems.Extensively classroom-tested to make sure an available presentation, Jet Single-Time Lagrange Geometry and Its purposes is a wonderful ebook for classes on differential geometry, relativity idea, and mathematical versions on the graduate point. The ebook additionally serves as a very good reference for researchers, execs, and teachers in physics, biology, arithmetic, and economics who wish to research extra approximately model-providing geometric buildings.

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The curvature of a normal iV-linear connection in the Miron's case is characterized only by three effective adapted components (see [56], p. 25). Remark 60. For the Berwald Γ-linear connection BT, associated to the metrics h\\ (t) and φ^(χ), all local curvature d-tensors vanish, except n(')(i) — _ pi¿ fe —_ ml (i)(*)jfc J yfc' where 9Ί' -fc(x) are the local curvature tensors of the semi-Riemannian metric

Remark 54. Theorem 53 implies that an /i-normal Γ-linear connection V is determined by four effective local components (instead of two effective local components for a normal Af-linear connection in the Miron's case [56], p. 21), namely, VT = ( « > , , G^L^C™). 3). 1 The canonical Berwald Γ-linear connection associated to the pair of metrics (/iii(i), yjjj(x)) is an /i-normal Γ-linear connection, defined by the local components B f = ( « ! , 0,7$, 0 ) . The study of adapted components of the torsion tensor T and curvature tensor R of an arbitrary Γ-linear connection V on E — J 1 (R, M) was completely done in the preceding Chapter, where we proved that the torsion tensor T is determined by ten effective local d-tensors, while the curvature tensor R is determined by fifteen effective local d-tensors.

3), is called an h-normal Γ-linear connection on E — J1 (R, M). Remark 52. The condition VJ^ = 0 is equivalent with the local equalities jd) _ Q (l)lj/1 - U' J _ A - U> T(») J (l)lj\k J T(») id) _ Q (l)ljl(fc) - U ' where "/i," "|fc," and "|L\" represent the R-horizontal, M-horizontal, and vertical local covariant derivatives produced by the Γ-linear connection VT. In this context, we can prove the following important local geometrical result: Theorem 53. 3) are a direct consequence of the definition of an /i-normal Γ-linear connection V.

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