Download Jet Single-Time Lagrange Geometry and Its Applications by Vladimir Balan PDF
By Vladimir Balan
Develops the idea 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 merits of jet single-time Lagrange geometry for geometrical modeling. With complete chapters that define themes ranging in complexity from simple to complicated, the booklet explores present and rising purposes throughout a vast variety of fields, together with arithmetic, theoretical and atmospheric physics, economics, and theoretical biology.The authors start through proposing easy theoretical options that function the basis for realizing how and why the mentioned idea works. Subusequent chapters examine the geometrical and actual elements of jet relativistic time-dependent Lagrange geometry to the classical time-dependent Lagrange geometry. a set of jet geometrical items also are tested comparable to d-tensors, relativistic time-dependent semisprays, harmonic curves, and nonlinear connections. quite a few purposes, together with the gravitational thought constructed by means of either the Berwald-Mo?r metric and the Chernov metric, also are presented.Throughout the booklet, the authors provide a variety of examples that illustrate how the idea is placed into perform, and so they current a number of functions during which the recommendations of first-order usual differential equation platforms are considered as harmonic curves on 1-jet areas. additionally, various possibilities are supplied for readers to achieve ability in employing jet single-time Lagrange geometry to unravel quite a lot of problems.Extensively classroom-tested to make sure an obtainable presentation, Jet Single-Time Lagrange Geometry and Its functions is a wonderful publication for classes on differential geometry, relativity concept, and mathematical versions on the graduate point. The ebook additionally serves as a great reference for researchers, pros, and teachers in physics, biology, arithmetic, and economics who wish to research extra approximately model-providing geometric constructions.
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Extra info for Jet Single-Time Lagrange Geometry and Its Applications
Example text
9) attached to the pair of metrics (hn(t),ipij(x)) are exactly the affine maps between the manifolds (R, Λιι(ί)) and (M, φ^(χ)). This page intentionally left blank CHAPTER 2 DEFLECTION d-TENSOR IDENTITIES IN THE RELATIVISTIC TIME-DEPENDENT LAGRANGE GEOMETRY This Chapter provides an extension of the Miron-Anastasiei geometrical framework from [55] and describes on the 1-jet space J X (R, M) the adapted components of the Γ-linear connections, together with their d-torsions and d-curvatures. For an arbitrary Γ-linear connection, the local Ricci identities, together with their corresponding nonmetrical deflection d-tensor identities, are also determined.
5. νΥι:ωΒ = -TBFCUJF, 6. F^BC' (2-4) where " :D " represents one from the local covariant derivatives "/χ," "y" or "|[ -j" produced by the Γ-linear connection V. Taking into account that the indices A, B, C,... 4) imply the required Ricci identities. Now, let us consider the canonical Liouville d-tensor field dy\ yi dy\ Definition 48. The distinguished tensors defined by the local components Γ>Μ (Di u — c^ ~ ^(i)/i' η ^ {i)j u _ r*W - ^(l)b' j(*)(i) _ ri(t)|(i) (i)(j) - ^(i)l(j) a -. n ^z·^ are called the nonmetrical deflection d-tensors attached to the T-linear connection V on the 1 -jet space E = J ^ R , M).
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.