Euler-Lagrangian Dynamical Systems with Respect to Horizontal and Vertical Lifts on Tangent Bundle

The differential geometry and mahthematical physics has lots of applications. The Euler-Lagrangian mechanics are very important tools for differential geometry, classical and analytical machanics. There are many studies about Euler-Lagrangian dynamics, mechanics, formalisms, systems and equations. The classic mechanics firstly introduced by J. L. Lagrange in 1788. Because of the investigation of tensorial structures on manifolds and extension by using the lifts to the tangent or cotangent bundle, it is possible to generalize to differentiable structures on any space (resp. manifold) to extended spaces (resp. extended manifolds) [5, 6, 9]. In this study, the Euler-Lagrangian theories, which are mathematical models of mechanical systems are structured on the horizontal and the vertical lifts of an almost complex structure in tangent bundle TM. In the end, the geometrical and physical results related to Euler-Lagrangian dynamical systems are concluded.