Weyl-Euler-Lagrange Equations of Motion on Flat Manifold

This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.


Introduction
Euler-Lagrangian (analogues) mechanics are very important tools for differential geometry and analytical mechanics.They have a simple method to describe the model for mechanical systems.The models for mechanical systems are related.Studies in the literature about the Weyl manifolds are given as follows.Liu and Jun expand electronic origins, molecular dynamics simulations, computational nanomechanics, and multiscale modelling of materials fields [1].Tekkoyun and Yayli examined generalized-quaternionic Kählerian analogue of Lagrangian and Hamiltonian mechanical systems [2].The study given in [3] has the particular purpose to examine the discussion Weyl and Einstein had over Weyl's 1918 unified field theory for reasons such as the epistemological implications.Kasap and Tekkoyun investigated Lagrangian and Hamiltonian formalism for mechanical systems using para-/pseudo-Kähler manifolds, representing an interesting multidisciplinary field of research [4].Kasap obtained the Weyl-Euler-Lagrange and the Weyl-Hamilton equations on R 2 which is a model of tangent manifolds of constant sectional curvature [5].Kapovich demonstrated an existence theorem for flat conformal structures on finite-sheeted coverings over a wide class of Haken manifolds [6].Schwartz accepted asymptotically Riemannian manifolds with nonnegative scalar curvature [7].Kulkarni identified some new examples of conformally flat manifolds [8].Dotti and Miatello intend to find out the real cohomology ring of low dimensional compact flat manifolds endowed with one of these special structures [9].Szczepanski presented a list of sixdimensional Kähler manifolds and he submitted an example of eight-dimensional Kähler manifold with finite group [10].Bartnik showed that the mass of an asymptotically flat manifold is a geometric invariant [11].González considered complete, locally conformally flat metrics defined on a domain Ω ⊂   [12].Akbulut and Kalafat established infinite families of nonsimply connected locally conformally flat (LCF) 4-manifold realizing rich topological types [13].Zhu suggested that it is to give a classification of complete locally conformally flat manifolds of nonnegative Ricci curvature [14].Abood studied this tensor on general class almost Hermitian manifold by using a new methodology which is called an adjoint -structure space [15].K. Olszak and Z. Olszak proposed paraquaternionic analogy of these ideas applied to conformally flat almost pseudo-Kählerian as well as almost para-Kählerian manifolds [16].Upadhyay studied bounding question for almost manifolds by looking at the equivalent description of them as infranil manifolds Γ \  ⋊ / [17].

Preliminaries
Definition 1.With respect to tangent space given any point  ∈ , it has a tangent space    isometric to R  .If one has a metric (inner-product) in this space ⟨, ⟩  :    ×     → Theorem 4. The integrability of the almost complex structure implies a relation in the curvature.Let { 1 ,  1 ,  2 ,  2 ,  3 ,  3 } be coordinates on R 6 with the standard flat metric: (see [18]).
where Ψ > 0 is a smooth positive function.An equivalence class of such metrics is known as a conformal metric or conformal class and a manifold with a conformal structure is called a conformal manifold [21].

Weyl Geometry
Conformal transformation for use in curved lengths has been revealed.The linear distance between two points can be found easily by Riemann metric.Many scientists have used the Riemann metric.Einstein was one of the first to study this field.Einstein discovered the Riemannian geometry and successfully used it to describe general relativity in the 1910 that is actually a classical theory for gravitation.But the universe is really completely not like Riemannian geometry.Each path between two points is not always linear.Also, orbits of moving objects may change during movement.So, each two points in space may not be linear geodesic.Then, a method is required for converting nonlinear distance to linear distance.Weyl introduced a metric with a conformal transformation in 1918.The basic concepts related to the topic are listed below [22][23][24].
Definition 11.Two Riemann metrics  1 and  2 on  are said to be conformally equivalent iff there exists a smooth function  :  → R with In this case,  1 ∼  (2) (, ) = (  , ) iff  =  − .So where  is a conformal structure.Note that a Riemann metric  and a one-form  determine a Weyl structure; namely,  : , where  is the equivalence class of  and (  ) =  − .
Theorem 19.Let ∇ be a torsion-free connection on the tangent bundle of  and  ≥ 6.If (, , ∇, ) is a Kähler-Weyl structure, then the associated Weyl structure is trivial; that is, there is a conformally equivalent metric g =  2  so that (, g, ) is Kähler and so that ∇ = ∇ g [25][26][27].
Definition 20.Weyl curvature tensor is a measure of the curvature of spacetime or a pseudo-Riemannian manifold.
Like the Riemannian curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic.
Definition 21.Weyl transformation is a local rescaling of the metric tensor:   () →  −2()   () which produces another metric in the same conformal class.A theory or an expression invariant under this transformation is called conformally invariant, or is said to possess Weyl symmetry.The Weyl symmetry is an important symmetry in conformal field theory.

Complex Structures on Conformally Flat Manifold
In this section, Weyl structures on flat manifolds will be transferred to the mechanical system.Thus, the time-dependent Euler-Lagrange partial equations of motion of the dynamic system will be found.A flat manifold is something that locally looks like Euclidean space in terms of distances and angles.
The basic example is Euclidean space with the usual metric  2 = ∑   2  .Any point on a flat manifold has a neighborhood isometric to a neighborhood in Euclidean space.A flat manifold is locally Euclidean in terms of distances and angles and merely topologically locally Euclidean, as all manifolds are.The simplest nontrivial examples occur as surfaces in four-dimensional space as the flat torus is a flat manifold.It is the image of (, ) = (cos , sin , cos , sin ).
Example 22.It vanishes if and only if  is an integrable almost complex structure; that is, given any point  ∈ , there exist local coordinates (  ,   ),  = 1, 2, 3, centered at , following structures taken from The above structures (7) have been taken from [28].We will use   = /  and   = /  .
The Weyl tensor differs from the Riemannian curvature tensor in that it does not convey information on how the volume of the body changes.In dimensions 2 and 3 the Weyl curvature tensor vanishes identically.Also, the Weyl curvature is generally nonzero for dimensions ≥4.If the Weyl tensor vanishes in dimension ≥4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor.This fact was a key component for gravitation and general relativity [29].
Proposition 23.If we extend (7) by means of conformal structure [19,30], Theorem 19 and Definition 21, we can give equations as follows: such that they are base structures for Weyl-Euler-Lagrange equations, where  is a conformal complex structure to be similar to an integrable almost complex  given in (7).From now on, we continue our studies thinking of the (, , ∇, ) instead of

Advances in Mathematical Physics
Weyl manifolds (, , ∇, ).Now,  denotes the structure of the holomorphic property: and in similar manner it is shown that As can be seen from ( 9) and ( 10)  2 = − are the complex structures.

Euler-Lagrange Dynamics Equations
Definition 24 (see [31][32][33]).Let  be an -dimensional manifold and  its tangent bundle with canonical projection   :  → . is called the phase space of velocities of the base manifold .Let  :  → R be a differentiable function on  and it is called the Lagrangian function.We consider closed 2-form on  and Φ  = −d  .Consider the equation where the semispray  is a vector field.Also, i is a reducing function and i  Φ  = Φ  ().We will see that, for motion in a potential,   = V() −  is an energy function ( =  −  = (1/2)V 2 − ℎ, kinetic-potential energies) and V =  a Liouville vector field.Here,   denotes the differential of .We will see that (11)
Proposition 25.Let (  ,   ) be coordinate functions.Also, on (, , ∇, ), let  be the vector field determined by  = ∑ 3 =1 (  (/  ) +   (/  )).Then the vector field defined by is thought to be Weyl-Liouville vector field on conformally flat manifold (, , ∇, ).Φ  = −d   is the closed 2-form given by (11) =1 (  (/  )+  (/  )).Also, the vertical differentiation d  is given where  is the usual exterior derivation.Then, there is the following result.We can obtain Weyl-Euler-Lagrange equations for classical and quantum mechanics on conformally flat manifold (, , ∇, ).We get the equations given by Also, and then we find Moreover, the energy function of system is and the differential of   is Using (11), we get first equations as follows: From here If we think of the curve , for all equations, as an integral curve of , that is, () = (/)(), we find the following equations: such that the differential equations ( 21) are named conformal Euler-Lagrange equations on conformally flat manifold which is shown in the form of (, , ∇, ).Also, therefore, the triple (, Φ  , ) is called a conformal-Lagrangian mechanical system on (, , ∇, ).

Weyl-Euler-Lagrangian Equations for Conservative Dynamical Systems
Proposition 26.We choose  = i  ,  = Φ  , and  = 2 at (11) and, by considering (4), we can write Weyl-Lagrangian dynamic equation as follows: The second part (11), according to the law of conservation of energy [32] and these differential equations ( 24) are named Weyl-Euler-Lagrange equations for conservative dynamical systems which are constructed on conformally flat manifold (, , ∇, , ) and therefore the triple (, Φ  , ) is called a Weyl-Lagrangian mechanical system.