Lagrangian Dynamical Systems with Three Para-complex Structures

This paper aims to present Lagrangian Dynamical systems formalism for mechanical systems using Three ParaComplex Structures, which represent an interesting multidisciplinary field of research. As a result of this study, partial differential equations will be obtained for movement of objects in space and solutions of these equations. In this study, some geometrical, relativistic, mechanical, and physical results related to Three ParaComplex Structures mechanical systems broad applications in mathematical physics, geometrical optics, classical mechanics, analytical mechanics, mechanical systems, thermodynamics, geometric quantization and applied mathematics such as control theory.


Introduction
The geometric study of dynamical systems is an important chapter of contemporary mathematics due to its applications in Mechanics, Theoretical Physics. If M is a differentiable manifold that corresponds to the configuration space, a dynamical system can be locally given by a system of ordinary differential equations of the form ; , which are called equations of evolution. Globally, a dynamical system is givenby a vector field X on the manifold whose integral curves, aregiven by the equations of evolution, .
The theory of dynamicalsystems dealwith the integration of such systems.
Mehmet and Murat Sari obtained On Para -Euler Lagrange and Para Hamiltonian Equations and constrained para complex Mechanical Equations [1].
Tekkoyun submitted paracomplex analogue of the Euler-Lagrange equations was obtained in the framework of para-Kahlerian manifold and the geometric results on a paracomplex mechanical systems were found [2].
Kasap and Tekkoyun obtained Lagrangian and Hamiltonian formalism for mechanical systems using para/ pseudo-Kahler manifolds, representing an interesting multidisciplinary. field of research. Also, the geometrical, relativistical, mechanical and physical results related to para/ pseudo-Kahler mechanical systems were given, too [3].
Oguzhan and Kasap submitted Mechanical Equations with Two Almost Complex Structures on Symplectic Geometry, using two complex structures, examined mechanical systems on Symplectic geometry [4].
In this paper, we study dynamical systems with Three Almost para Complex Structures. After Introduction in Section 1, we consider Historical Background paper basic. Section 2 deals with the study paracomplex Structures. Section 3 is devoted to study Lagrangian Dynamics.

Preliminaries
In this preliminary chapter, we recall basic definitions, results and formulas which we shall use in the subsequent chapters of the paper. Most of material included in this chapter occurs in standard literatures namely.

Definition 2.1. [6]
An almost product structure J on a tangent bundle T of m-real dimensional configurationmanifold is a (1, 1) tensor field J on T such that I I. Here, the pair T , J is calledan almost product manifold.

Definition 2.2. [8]
An almost Para -complex structure on amanifold is a differentiable map I: T T on the tangent bundle of such that preserves each fiber, A manifold with affixed almost para -complex structure is called an almost para -complex manifold.
And the dual convector fields Suppose that , , , , , ! " be a real coordinate system on (ℳ, #). Then we denote by.
If I is defined as a Para complex manifoldℳ then = I ∘ I = 1 Proof.

Lagrangian Dynamical Systems
In this section we introduce the concept of Lagrangian Dynamical Systems. We start by the following definition.
Definition 3.1. [5] A Lagrangian function vector field X onℳ is a smooth function L ∶ Tℳ → R such that.
Let D be the vector field by.