On a New Conformal Euler-Lagrangian Equations on Para Quaternionic 𝐾�𝐾�𝑎�𝑎�̈ℎ𝑙�𝑙�𝑙�𝑙�𝑙�𝑙� Manifold

Abstract

Today, many branches of science are into our lives. One the branches is mathematics that has multiple applications. In particular , differential geometry and mathematical physics have a lots of different applications. One of them are on geodesics. Geodesics are known the shortest route between two points. Time-dependent equations of geodesics can be easily found with the help of the Euler-Lagrange equations. We can say that differential geometry provides a good working area for studying Lagrangians of classical mechanics and field theory. The dynamic equation for moving bodies is obtained for Lagrangian mechanic. These dynamic equation is illustrated as follows: Lagrange Dynamics Equation [1,2,3]: let M be an n-dimensional manifold and TM its tangent bundle with canonical projection 𝑇𝑇𝑇𝑇: 𝑇𝑇𝑇𝑇 → 𝑀𝑀 is called the phase space of velocities of the base manifold M. Let 𝐿𝐿: 𝑇𝑇𝑇𝑇 → 𝑅𝑅 be differentiable function on TM called the Lagrangian function. We consider the closed 2-form on TM given by Φ𝐿𝐿 = 𝑑𝑑𝑑𝑑𝐽𝐽𝐿𝐿 ( if 𝐽𝐽2 = −𝐼𝐼 ,𝐽𝐽 is a complex structure and if 𝐽𝐽2 = 𝐼𝐼 ,𝐽𝐽 is a paracomplex structures, 𝑇𝑇𝑟𝑟 (𝐽𝐽) = 0) Consider the equation: 𝑖𝑖𝑋𝑋Φ𝐿𝐿 = 𝑑𝑑𝐸𝐸𝐿𝐿

→ Then 𝑋𝑋 is a vector field, we shall see that (1) under a certain condition on 𝑋𝑋 is the intrinsical expression of the Euler-Lagrange equations of motion. This equation is named as Lagrange dynamical equation. We shall see that for motion in potential, 𝐸𝐸𝐿𝐿 = 𝑉𝑉(𝐿𝐿) − 𝐿𝐿 is an energy function and 𝑉𝑉 = 𝐽𝐽(𝑋𝑋) a Liouville vector field. Here 𝑑𝑑𝐸𝐸𝐿𝐿 denotes the differential of 𝐸𝐸. The triple (𝑇𝑇𝑇𝑇, Φ𝐿𝐿, 𝑋𝑋) is known as Lagrangian system on