The aim of this article is firstly to improve time-dependent Lagrangian energy equations using the super jet bundles on supermanifolds. Later, we adapted this study to the graph bundle. Thus, we created a graph bundle by examining the graph manifold structure in superspace. The geometric structures obtained for the mechanical energy system with superbundle coordinates were reexamined with the graph bundle coordinates. Thus, we were able to calculate the energy that occurs during the motion of a particle when we examine this motion with graph points. The supercoordinates on the superbundle structure of supermanifolds have been given for body and soul and also even and odd dimensions. We have given the geometric interpretation of this property in coordinates for the movement on graph points. Lagrangian energy equations have been applied to the presented example, and the advantage of examining the movement with graph points was presented. In this article, we will use the graph theory to determine the optimal motion, velocity, and energy of the particle, due to graph points. This study showed a physical application and interpretation of supervelocity and supertime dimensions in super-Lagrangian energy equations utilizing graph theory.

中文翻译：

---

具有图集的超流形上的时变拉格朗日能量系统

本文的目的是首先使用超流形上的超级射束来改进与时间有关的拉格朗日能量方程。后来，我们使这项研究适应图捆绑。因此，我们通过检查超空间中的图流形结构来创建图束。用图束坐标重新检查为具有超束坐标的机械能系统获得的几何结构。因此，当我们使用图形点检查运动时，我们能够计算出在运动期间发生的能量。超流形的超束结构的超坐标已经给出了身体和灵魂，以及偶数和奇数尺寸。我们在坐标上针对此图形点的运动给出了此属性的几何解释。拉格朗日能量方程已被应用于所提出的示例，并提出了利用图形点检查运动的优势。在本文中，我们将使用图论来确定由于图点而导致的粒子的最佳运动，速度和能量。这项研究显示了利用图论在超拉格朗日能量方程中超速度和超时维的物理应用和解释。