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Mechanical Equations of Dynamic Systems on Weyl-Finsler Manifolds

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Mechanical Equations of Dynamic Systems on Weyl-Finsler Manifolds Open Access Journal of Mathematical and Statistical Analysis RESEARCH ARTICLE ISSN: 2642-018X Mechanical Equations of Dynamic Systems on Weyl-Finsler Manifolds Zeki Kasap* Faculty of Education, Department of Elementary Education, Pamukkale University, Kinikli Campus, Denizli, Turkey *Corresponding author: Zeki Kasap, Faculty of Education, Department of Elementary Education, Pamukkale University, Kinikli Campus, Denizli, Turkey, E-mail: [email protected]; [email protected] Citation: Zeki Kasap (2018) Mechanical Equations of Dynamic Systems on Weyl-Finsler Manifolds. J Math Stat Anal 1: 107 Article history: Received: 21 June 2018, Accepted: 26 October 2018, Published: 28 October 2018 Abstract Finsler geometry is an alternative approach to geometrization of fields and its fundamental idea can be traced of Riemann geometry so that Finsler geometry is a nearest generalization of Riemannian metric geometry. Finsler geometry was first applied in gravitational theory and this application lead to corrections to observational results predicted by general relativity. It’s well known that there are various approaches to Finsler formulation of the general relativity and gauge field theories in the spacetime. Euler-Lagrange and Hamilton equations often used for modeling of mechanical systems. In this study, we will present Euler-Lagrange and Hamilton equations often used for modeling of mechanical systems of motion on Finsler manifolds such that they represent the dynamic state of the mechanical system. Also, implicit solutions of the differential equations found in this study are solved by Maple computation program. Keywords: Weyl Manifold; Lagrangian; Hamiltonian; Conformal Structure; Dynamics Equation; Finsler Manifold AMS: 30C20; 34B20; 58B20; 70H03; 70H05; 70E55 Introduction A classical field theory explains the study of how one or more physical fields interact with matter which is used quantum and classical mechanics of physics branches. Also, a unified theory and it’s the metric has been thought as the gravitational potential, as in general relativity, and the corresponding forms are thought as the electromagnetic potentials. Weyl’s unified theory and it’s the metric has been thought as the gravitational potential, as in general relativity, and the corresponding forms are thought as the electromagnetic potentials. Euler-(Lagrangian) and Hamiltonian models arise to be a very important tool and they present a simple method to describe the model for orbits of moving objects in the electromagnetic field. There are many applications of differential geometry or mathematical physics. These applications are used in many areas of disciplines. There are many studies about Lagrangian and Hamiltonian dynamics, mechanics, formalisms, systems, equations and Finsler geometry [1]. Tekkoyun and Celik presented a new analogue of Euler-Lagrange and Hamilton equations on an almost Kähler model of a Finsler manifold [2]. Tekkoyun and Yayli presented generalized-quaternionic Kähler analogue of Lagrangian and Hamiltonian mechanical systems [3]. Wu established a relative volume comparison theorem for minimal volume form of Finsler manifolds under integral Ricci curvature bound [4]. Vries shown that the Hamiltonian and Lagrangian motion equations have a very simple interpretation in relativistic quantum mechanics [5]. Udriste and Neagu presented the basic properties of the scalar product along a curve and they investigated the variational formulae for the p-energy functional on a Finsler manifold [6]. Bercu showed the gradient method on Finsler manifold as to how to use the direction y for obtaining a suitable descent algorithm [7]. Szilasi and Tóth deduced consequences on vector field on the underlying manifold of a Finsler structure having one or two of the mentioned geometric properties [8]. Tayebi and Peyghan constructed a new class of Finsler metrics which is an extension of the class of Berwald metrics [9]. Bejancu constructed the transversal vector bundle of a coisotropic submanifolds of pseudo-Finsler manifold and obtained all structure equations of the degenerate immersion [10]. Lovas and Szilasi propose a new and complete proof of the theorem, discovered by Detlef Laugwitz: complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces [11]. Abate and Patrizio showed that a complex Finsler metric of constant holomorphic sectional curvature-4 satisfying the given symmetry condition on the curvature is necessarily the Kobayashi metric [12]. Almost contact Finsler structures on vector bundle are defined by Yaliniz and Caliskan and the condition of normality in terms of the Nijenhuis torsion N{φ} of almost contact Finsler structure is obtained [13]. Vacaru considered some classes of exact solutions instring and Einstein gravity modelling Lagrange-Finsler structures with solitonic pp--waves and speculate on their physical meaning [14]. Kasap demonstrated Weyl-Euler-Lagrange and Weyl-Hamilton equations on R{n}²ⁿ which is a model of ScholArena | www.scholarena.com Volume 1 | Issue 1 2 J Math Stat Anal tangent manifolds of constant w-sectional curvature [15]. Kasap submitted Weyl-Euler-Lagrange equations of motion on flat manifold [16]. Miron and Anastasiei and their other colleagues there are summarized a number of results on almost Kähler- Lagrange-Finsler/Hamilton-Cartan geometries and generalizations with applications in mechanics [17]. Preliminaries Definition 1: Let (M,F) be a connected n-dimensional Finsler manifold whose fundamental function verifies F:TM→ R the following axioms: (F1) F(x,y)>0; ∀xεM, ∀y≠0. (F2) F(x,λy)=|λ|F(x,y); ∀λεR, ∀(x,y)εTM. i j ∀ (F3) the fundamental tensor gij(x,y)=(1/2)((∂²F²)/(∂y ∂y )) is positive definite; xεM, (F1) F( xy,) > 0; ∀ xMε , ∀≠ y 0. (F2) F( x,λ y) = λ F( xy ,) ; ∀∀ λε R ,( xy ,) ε TM . ij (F3) the fundamental tensor gij ( x, y) =( 1/2)(( ∂ ²F ²) /( ∂∂ y y)) is positive definite;∀ xε M , c (F4) F is C at every point (x,y)εTM with y≠0 and continuous at every (x,0)εTM. Then, the absolute Finsler energy is F²(x,y)=gij(x,y) yiyj. ∞ Letc: [ ab,] → MbeaC Xt= Xii t ∂∂/ x | =ii ∂∂ ┊ Definition 2: regular curve on M. For any two vector fields ( ) ( )( ( )) ct()┊, Yt( ) Y( t)( / ( x)) | ct( ) along the curve c . g( XY,,)(( ct( ) = g ct( ) Ct( ) ) XYij and the scalar product ij along the curve c. Definition 3: Let M be a differentiable manifold of dimension (2n+1) and suppose J is a differentiable vector bundle isomorphism J:TM→TM such that J:TM→TM is an almost complex structure for TM. Definition 4: An almost complex structure J on M assigns to each pЄM a linear map Jp:TpM→TpM that is smooth in p and satisfies Jp²=-Id for all p. The pair (M,J) is called an almost complex manifold. Any complex manifold M is also an almost complex manifold. In three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) can be written as [18]. rx=i + y jk + z = x( ∂∂ /( x)) + y( ∂∂ /( y)) +∂∂ z( /( z)) . (1) Lemma 1: Let M be a smooth manifold. If M admits a complex structure A, then M admits an almost complex structure J. Let dimCM=m and (Z,U) be any holomorphic chart inducing a coordinate frame ∂x₁,∂y₁,...,∂xm,∂ym. Then J is given locally as Jp()∂=∂ xii| p┊┊ y| p, Jp(∂ yii |) p┊┊ = −∂ x| p, (2) Where 1≤i≤m and pЄU [19] Suppose M is a smooth manifold. Recall that a smooth curve in M is a smooth map γ :IM→ , where I is an interval in R. For any aЄI, the tangent vector of γ at the point γ(a) is γγ= = ()a(( d) /( dt))( a) dγ (a) (( d/( dt))) (3) where d/dt is the standard coordinate tangent vector of R. Let X is a smooth vector field on M. We say that a smooth curve γ: I→M is an integral curve of X if for any tЄI, γ tX= . ( ) γ ()t (4) Geodesics in Finsler Spaces A Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasisimetric space in which the length of any rectifiable curve γ:[a,b]→M is given by the length functional b L[γ ]= ∫ F ( γγ ( t ), ( t )) dt (5) a F(X,X) Where is a Minkowski norm on each tangent space TxM. Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. ScholArena | www.scholarena.com Volume 1 | Issue 1 J Math Stat Anal 3 This means that the norm on each tangent space is not necessarily induced by an inner product (metric tensor). Due to the homogeneity of F the length (5) of a differentiable curve γ:[a,b]→M in M is invariant under positively oriented reparametrizations. A constant speed curve γ is a geodesic of a Finsler manifold if its short enough segments γ |[]cd, ┊ are length-minimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional b 2 E[γ ]= (1/ 2)∫ F ( γγ ( t ), ( t )) dt (6) a in the sense that its functional derivative vanishes among differentiable curves γ:[a,b]→M with fixed endpoints γ(a)=x and γ(b)=y. A Finsler manifold is a differentiable manifold M together with a Finsler function F defined on the tangent bundle of M so that for all tangent vectors v. For each tangent vector v, the Hessian of F² at v is positive definite. Here the Hessian of F² at v is the symmetric bilinear form gv( X, Y ) =( 1/ 2)(( ∂ ²) / ( ∂∂s t)┊) F( v + sX+ tY )² |st= =0 , (7) Also known as the fundamental tensor of F at v. Strong convexity of F² implies the subadditivity with a strict inequality if u/F(u)≠v/F(v). If F² is strongly convex, then F is a Murkowski norm on each tangent space. Gauge Theory and Conformal Weyl Geometry Hermann Weyl (1885-1955) made many fundamental and important contributions to physics.
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