#### Abstract

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.

#### 1. Introduction

Many of the problems based on Lagrangian energy equations in classical mechanics may be solved using Euclidian space. It is well known that modern differential geometry has an important role too. Many of these problems have not been calculated in superspace before. One reason may be the difficulty that the metric structure of superspace differs from Euclidean space. Using jet bundles and working with bundle coordinates is one of the most convenient ways to obtain the time-dependent Lagrangian energy equations. Therefore, in this study, supercoordinate structure will be formed on jet bundles and superenergy equations will be obtained by using this.

Mathematicians, working in superspace, assume that there is a natural phenomenon with supergeometry and supersymmetry for explaining physical phenomena occurring in even- and odd-dimensional Euclid space. Thus, we prefer to solve Lagrangian energy equations on the superspace, which is based on a jet bundle structure. Inclusion of time dimension for solving Lagrangian energy equations on superspace is an important parameter that improves the super-Lagrangian system for which we propose to take time derivative coordinates on the jet bundle.

On the other hand, graph theory is the most suitable method for planning a movement that completes a path as soon as possible. In this study, we will compare the graph bundle structure and the bundle structure that we use in calculating energy equations in the mechanical system. We will combine these two bundle structures and adapt them in the superspace. We will be able to define the motion depending on time with graph points and also form a graph vector field with graph vectors. Thus, we will combine the geometric concepts used in calculating Lagrangian energy equations with graph theory, and we will calculate the energy equation with a completely new approach. A study has been done on the graphs of the logarithmic spiral curve in [1]. In [2], using the graph theory, the study of Euler–Lagrange energy structure in neural networks has been examined. Cangül et al. [3, 4] is a basic resource on graph theory. Graph manifold structure is defined in [5–7]. Topological properties of the graph manifold are investigated in [8]. Graph bundle structure is examined in [9].

Aycan [10] proved the Lagrange energy equations on the jet bundle structure containing the time dimension. Also, in [11], Lagrangian energy equations are developed by forming jet bundles on a complex space. Lagrangian equations are solved with real bundles by [12–14]. Mechanical systems with time parameter were investigated in [15–17]. The other studies [18–24] showed the fundamental of supergeometric structures. But none of them could be solved with superjet bundles that include the time dimension in the superspace. Thus, the main contribution of this study is to obtain super-Lagrange energy equations with bundle structures structured with body and soul and also even and odd dimensions, and we will form it with a graph bundle with time dimension. The advantage of this solution method is that the energy equations found in superspace can be projected or comparable with energy equations in real space.

In previous studies, super-Lagrangian equations in supermanifolds have not been studied for superderivative coordinates. There for this study improves this equation in superderivative coordinates with body and soul and also even and odd dimensions.

A brief introduction of Lagrangian systems is given in the following way.

If *M* is an *m*-dimensional configuration manifold and is a regular Lagrangian function, then there is a unique vector field on TM and is a 2-form on TM, such thatwhere is Lagrangian energy associated with *L* [12, 15]. The Euler–Lagrange vector field is a semispray or second-order differential equation on *M* since its integral curves are the solutions of the Euler–Lagrange equations [12, 15]. The triple (TM, , *L*) is called the Lagrangian system on the tangent bundle TM [12]. The coordinate system on is . For created to time-dependent Lagrange systems are used jet bundles. When studying energy systems, it will be very convenient for obtaining energy equations with the jet bundle structure because the motion depends on time and therefore to take the time as a coordinate. At the same time, the jet bundle is the velocity space of the manifold *M*. is isomorphic to . The coordinate system on is . On jet bundles, then there is a unique vector field (or semispray) and a 2-form called as Poincaré–Cartan 2-form again so that they provide the following equations:

Here, is the Lagrangian function. The triple (, , ) is called the time-dependent Lagrangian system. The Euler–Lagrange vector field is locally expressed as

We have to define Poincaré–Cartan 1-form to get Poincaré–Cartan 2-form. The Poincaré–Cartan 1-form on associated with *L* is

The Poincaré–Cartan 2-form associated with *L* is

When solving equation (2),is obtained. Equation (2) is called as time-dependent Euler–Lagrange energy equation. The same equation would be found if the solution is made in equation (1) because the Lagrange energy equation is similar. When we work in different spaces in our previous work, we have seen that the obtained Lagrange energy equation can be reduced to (6). In this study, we will obtain the energy equation by creating the necessary geometric structures to obtain the Lagrange energy equation in superspace and compare it with equation (6).

#### 2. Bundles on Superspace

In this section, we will define a bundle in superspace by using the bundle definitions in real space and graph theory. Then, we will form the geometric structures required for the solution of the mechanical system firstly with the jet bundle coordinates and then graph bundle coordinates. With the solution of this mechanical system, we will obtain the Lagrangian energy equation.

*Definition 1. *Let be a bundle. Here, is called the total space, is the projection, and is the base space. This bundle is denoted by or . For each point the subset of E is called as fiber over p. The set of fibers is shown as . So the bundle can also be called a fibrous manifold. *F* is the typical fiber of the bundle, and for the map , is called a trivialization.

The first jet manifold of is the set and denoted by or Here, is a section of If it is satisfying the condition *π*, then the set of all sections of will be denoted by . , if and and this is an equivalence relation and the equivalence classes containing are called the first jet at the *p*-point. These are denoted by . Also, the triple is the first jet bundle and denoted by . Here, is a surjective submersion. TM is the tangent bundle of the manifold *M*. (TM, , *M*) is named as tangent bundle. At the same time, is isomorphic to TM, , that is, the first jet is also a tangent vector [10].

Let be an adapted coordinate system on *E*, where. The induced coordinate system on is denoted by , where and new functions are known as derivative coordinates.

We will define the superbundle with these properties.

*Definition 2. *A superbundle is a triple where and are supermanifolds and is a surjective submersion. This bundle is denoted by or . The first superjet manifold of is denoted by or . It is the set Here, is a map, and is called as supersection of . If it satisfies the condition , then the set of all supersections of will be denoted by . The equivalence classes containing are called the first jet at the *p*-point and denoted by . Also, , if and , and this is an equivalence relation and the equivalence classes containing are called the first jet at the *p*-point. Then, the first superjet bundle is the triple and denoted by . is isomorphic to T, , that is, the first superjet is also a supertangent vector.

The induced coordinate system on will be obtained as follows.

In this paper, we accept the manifold as a supermanifold with *m* + *n* dimension, and the manifold is a supermanifold with *m*-dimension. Let us show the adapted coordinate system on is . Since this is a superspace coordinate, its open representation will be made as follows:where is the body part of and is the soul part of . On the other hand, has even and odd parity. We will show the even parity of as and the odd parity of as We show the adapted coordinate system on is . Similarly, the open representation of is taken as follows:where is the body part of and is the soul part of . On the other hand, has also even and odd parity. We will show the even parity of as and the odd parity of as For simplicity, the body indexes symbolize as *b* and the soul indexes symbolize as *s*. Also, we will obtain the derivative coordinates of superjet bundle as follows. The important point to be considered here is that the derivatives of the soul and body parts must be taken separately within themselves. We saw that the results were very different from what was desired in all the examinations we made by using the mixed derivatives for body and soul parts. So, the coordinates of are shown as follows:Here, we denote the derivative coordinates distinctly as follows: (derivative of body coordinates according to body coordinates). (derivative of soul coordinates according to soul coordinates). Here, has also even and odd parity too. We will show the even parity of as and the odd parity of as . Also, (derivative of even soul coordinates according to even soul coordinates). (derivative of odd soul coordinates according to odd soul coordinates).If these more explicit representations of derivative coordinates are used, we can write the adapted coordinate system of superjet bundle asIf we want to create a time-dependent jet bundle, we must take the real space as the base space. So, the triple we will use is and its coordinated system is . The adapted coordinate system on is , on is , and also on is :Accordingly, time-dependent jet bundle structure can also be formed in superspace. The supertime-dependent jet bundle structure will be shown as So, the coordinates of this bundle are shown as follows:The simple notation of the derivative coordinates here isNow we define graph manifold. *M* is a three-manifold. Class M consists of 3-dimensional, connected, closed, orientable manifold. Associated with every *M* is its graph *G*. The vertex set *V* of *G* is the set of maximal blocks in *M*, and the set *E* of the oriented edges of *G* can be identified with the set of boundary components of all maximal blocks. Namely, an edge is directed from a vertex *x* to a vertex *y*, if the boundary torus is attached in *M* where to the boundary torus where the minus sign means reserve edge orientation. The incompressible torus in *M* that results from gluing the boundary tori and will be denoted as . The set of the edges coming out from a vertex is denoted by , and if , then we write

*Definition 3. *Let *B* and *F* be graphs. A graph *G* is graph bundle with fiber *F* over the base graph *B* if there is a mapping which satisfies the following conditions:(1)It maps adjacent vertices of *G* to adjacent or identical vertices in *B*(2)The edges are mapped to edges or collapsed to a vertex(3)For each vertex and for each edge A mapping satisfying just the two conditions above is called a graph map. For a given graph *G*, there may be several mappings with the above properties. In such cases, we write (, ) to avoid confusion. Now we introduce an equivalence relation defined among the edges of a graph. With this relation can be recognized graph bundles [9].

An induced cycle of four vertices is called a chordless square. With this definition, we can define an auxiliary binary relation For any , we set if at least one of the following conditions is satisfied:(1)*e* and *f* are opposite edges of a chordless square(2)*e* and *f* are adjacent and there is no chordless square spanned on *e* and *f* [9]By , we denote the reflexive and transitive closure of Since is symmetric, is an equivalence relation. Any pair of adjacent edges which belong to distinct -equivalence classes spans a chordless square.

If we take graph manifolds instead of *E* and *M* manifolds in the bundle definition given in Definition 1, and a is also a graph map, a graph bundle is obtained when the conditions given in Definition 3 are provided. So, we can obtain the energy equation on a graph bundle. Now we have defined the graphs in superspace. So, we can create the supergraph bundle given by Definition 2. Since graph manifolds are usually taken 3-manifold, we will take the superspace in 3 dimensions in this study. In addition, since the graph manifolds are real, we will take the body parts of the supernumbers as real numbers in this study because if the body part of the number is real, it corresponds to a real number, and if the body part of the number is complex, it corresponds to a complex number.

A vertex in superspace will have the body part and the soul part as well as soul even and soul odd parts. Since graphs are formed by combining vertex and edges, we need to consider soul odd and soul even parts for vertex when defining graphs in superspace. Lines joining points called vertex in graph theory are called edges. In the mechanical system we have introduced to calculate the energy equation during the motion of a moving particle, these vertex points become the points that the object passes through during its motion. Edges also correspond to the paths formed by the movement of the object between these vertex points. It can also be regarded as a vector geometrically because it is a directional line segment. The combination of all the paths that can occur as the direction of motion changes again forms a graph in superspace. ( and are vertex and *E* is edge), and () is a graph. We will denote the parity with () for the graphs. In superspace, if , the number is odd, and if , the number is even. When defining geometric structures in superspace, the sign of odd parts must be negative. This is a necessity for the study to give correct results. Physically, it can be explained with direction. In geometry, the positive direction is counterclockwise direction, and the negative direction is on the contrary of it, that is, clockwise direction. In the graph manifold definition, we said that the minus sign is related to the orientation. Therefore, when defining the graph theory in superspace, the concept of direction should be determined by examining the soul odd and soul even parts of the vertices. A supernumber can consist of body, soul odd and soul even parts or just body, and soul even or soul odd parts. It is related to whether this number is an odd or an even number. Therefore, since the vertex points will be points in the 3-dimensional superspace we are working with, the movements between soul even and body parts of these vertex points will show the positive direction and the movements between the soul odd parts will show the negative direction. So, the movement between the soul even and body parts of vertex points is picturing counterclockwise direction and the movement between the soul odd parts of vertex points is picturing clockwise direction.

For example, let the vertex point only consist of body and soul even parts. Consider the movement from this point to another vertex point consisting of body and soul odd parts. Since the movement will first be from the body part to the body part, it will occur in the (+) direction (counterclockwise) and then it will happen in the (−) direction (clockwise) because it will be in the direction from the soul even part to the soul odd part. This can be seen in the graph manifold graph given in Figure 1.

get a graph on the *M* supermanifold. Here, is the set of vertexes and is the set of edges. Let any two subsets of be , alsoAccording to this, any open subset of is and an open subset of is . Here *m* is body dimension, is soul even dimension, and is soul odd dimension. The coordinate neighborhood system (atlas) can be obtained for the differential structure on a graph manifold in superspace by defining homeomorphisms on these sets as follows:to be one to one, so the pair is called a supergraph coordinate neighborhood in M. According to this,(1)(2)For to be , hence function to be differentiableThe collection that meets the above conditions is called a supergraph atlas in *M*.

Accordingly, we can picture the differential structure of a supergraph manifold in (Figure 1).

In Figure 1, is the open subset of vertexes that have a body part (*b*) is the open subset of vertexes that have a soul even part is the open subset of vertexes that have a soul odd part According to this,(1)In region 1, the *x* and *y* vertexes only consist of the body part, i.e., and (2)In region 2, the *x* and *y* vertexes consist of the body part and soul even part, i.e., and (3)In region 3, the *x* and *y* vertexes only consist of the soul even part, i.e., and (4)In region 4, the *x* and *y* vertexes consist of the body part and soul odd part, i.e., and (5)In region 5, the *x* and *y* vertexes consist of all parts for a supernumber, namely, the body part, soul even part, and soul odd part, i.e., and (6)In region 6, the *x* and *y* vertexes consist of the soul even part and soul odd part, i.e., and (7)In region 7, the *x* and *y* vertexes only consist of the soul odd part, i.e., and Now we define the homeomorphisms shown in Figure 1 as follows: where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge Here, thirty-one distinct homeomorphism maps can be defined between the body, soul even, and soul odd parts of the supersets, as follows: where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge where domain is the edge According to the transformations we have defined above, the edges obtained by combining the vertex on a graph manifold can be depicted as follows. Thus, the graph manifold can be shown in Figure 2. Considering each of the above *β* maps, edges are depicted linearly between vertexes with the same part, and edges between vertexes with different parts are depicted as curves because when combining vertexes with different parts, the coordinate order must be considered (body, soul even, and soul odd). These couplings are shown in Figure 2. The orientations in this graph will be determined as such: in vertexes, in the orientation from even parity to even parity, counterclockwise, that is, positive orientation is taken; in the orientation from even parity to odd parity or from odd parity to even parity, clockwise, that is, negative orientation is taken.

*Definition 4. *Let and be supergraph manifolds. A supergraph is a supergraph bundle with fiber over the base supergraph if there is a mapping which satisfies the following conditions:(1)It maps adjacent vertices of to adjacent or identical vertices in according to the orientation properties for the graph manifold (Figure 2)(2)Under the same conditions expressed above, the edges are mapped to edges or collapsed to a vertex(3)For each vertex and for each edge , A mapping satisfying just the two conditions above is called a supergraph map. We can also define this map as vertex-edge transformation, and in other words, the values created by the combination of each vertex we receive will turn into the edges formed by the combination of vertexes that these vertexes will correspond with this transformation. The subset of is also called supergraph fiber. is also a covering submersion.

The equivalence class defined among the edges can be defined as supergraph bundles. For any , we set if at least one of the following conditions is satisfied:(1)*e* and *f* are opposite edges between the body, soul even, or soul odd parts of the vertex that have a chordless square(2)*e* and *f* are adjacent according to directions in Figure 2 and there is no chordless square spanned on *e* and *f*Since is symmetric, it is an equivalence relation. Consequently, the triple (, ) is called a supergraph bundle. The dimensions of supergraph manifolds are as follows: boy boy Get the supergraph coordinate system on the superopen subset, where *x* and *y* are each edge; for instance, edge *x* is the combination of vertex and .If this proposition is true, then is called a supergraph adapted coordinate system.

The representation of this in graph theory is edge lines that can be orientable to the vertex from which they started, namely, point *x*. Since the first jets are identical to tangent vectors, the same identification is made for the first jets. Then, Thence, the necessary jet bundle to install time-dependent mechanical systems for supergraphs on the supergraph bundle (, ) will be (, ). Therefore, the coordinates of a graph jet bundle are expressed as follows, considering the coordinate systems we have given above:The coordinate *u* corresponds to an oriented edge *e* that can be written as for the initial vertex. For *x*, *y* vertexes, that is, *u* is also a directional line segment (geometrically a vector) from *x* to *y* which is corresponding to an edge. In graph theory, this coordinate corresponds to the derivative coordinate for the jet bundle. According to the derivative coordinate definition, *u* component can be expressed as follows, but for the graph manifold structure, the explanations given next to it will be taken into consideration:Here, as we explained in the bundle definition, the derivative of the body part with respect to the body part, the derivative of the soul even part with respect to the soul even part, and the derivative of the soul odd part with respect to the soul odd part will be taken. Mixed derivatives cannot be taken between parts. The components of edge coordinates (derivative coordinates) should also be expressed in this way. For example, when we get the following map, where domain is the edge , a coordinate of the graph jet bundle will take the form .

*Definition 5. * be a graph with finite vertex set *V* and edge set *E*. Given an oriented edge *e*, that can be written as for the initial vertex. Star is the tangent space of the graph manifold at a point *x*. Elements of this space are called tangent vectors [6].

Geometrically, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in *R*^{n}. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Consider a fixed point *X* and a moving point *P* on a curve. As point *P* moves toward *X*, the vector from *X* to *P* approaches the tangent vector at *X* (Figure 3). The line that contains the tangent vector is the tangent line. In this work, we will denote the tangent space of the graph manifold, that is, the set Star (*x*) with TS. Additionally, a tangent vector on the manifold *M* is a map . We have special tangent vectors (called the partial derivatives), , ( local coordinate system for *M* at *p*). tangent vectors system is the base for the tangent manifold TM.

The base for the graph tangent manifold TM is denoted in the same way, and this vector system is denoted as .

A vector bundle is a special class of fiber bundle in which the fiber is a vector space *V*. If is a bundle with fiber , to be a vector bundle, all of the fibers need to have a coherent vector space structure. Also, a vector bundle is a total space *E* along with a surjective map to a base manifold *B*. Any fiber is a vector space isomorphic to *V*. So, each point *x* of the manifold *M* corresponds to a vector. In graph theory, it is an orientable graph for each vertex, formed by vectors accepted as edges between these vertices. For example, in Figure 2, the system of edges formed between for a finite number of some paths *x* and *y* vertices on the graph manifold becomes a vector bundle.

While defining the geometric structures in Section 3 in graph theory, we will make use of the definitions and explanations given in this section.

#### 3. Lagrangian Mechanical Systems for Superspace with Superjet Bundle and Supergraph Bundle

In this section, we will first obtain Lagrangian mechanical systems with superjet bundle structure. Thus, we will have time-dependent Lagrange energy equations in superspace. Then we will construct these geometric structures for supergraph bundle. We will compare the time-dependent Lagrangian energy equation obtained from the solution of the mechanical system for graph bundle coordinates with the previous equation. Thus, the Lagrangian energy equation will be obtained for the graph bundle.

*Definition 6. *Let be a tensor field of type first-order covariant and first-order contravariant such that byWe will write this tensor field in coordinates as follows:This tensor field is named as super almost tangent structure and provides the condition

*Definition 7. *A semispray in superspace is a vector field over the total manifold and defined as follows:If we calculate the *J*-directional derivative of the semispray , we can obtain the Liouville vector field *V*:To set up and solve the Lagrangian mechanical system, we have to create Poincaré–Cartan 1-form and Poincaré–Cartan 2-form in superspace. Firstly, we must write the differential operator *d* with supercoordinate system as follows:Now let us form the Poincaré–Cartan 1-form first,By differentiating Poincaré–Cartan 1-form, we get Poincaré–Cartan 2-form as follows:

Theorem 1. *The Euler–Lagrange energy equation in superspace will be obtained as follows:*

*Proof. *Euler–Lagrange energy is obtained by solving the dynamical equation . Thus, let us set up and solve this dynamical equation using the geometric concepts we have defined above according to the supercoordinate system in superspace:By equalizing equation (29) to zero, then the following equations are obtained:Equation (30) is a system of nonlinear equations. For the solution of this nonlinear equation system, some special conditions are required. Among the different assumptions made, the most appropriate conditions for the general solution of the Lagrangian energy equation are determined as follows:We have seen that the general structure of the Lagrange equation shows similarities in different spaces in our previous articles. For this reason, the minus sign here has been accepted in order to create the similarity with the general energy equation, and another reason for this comes from the concept of parity in superspace. As a result of our study, it can be said that the choice of derivative coordinates on the coordinate system of the base manifold will be the most appropriate choice for the solution. These conditions are written in equation (30). Then, a common solution of all these equations is sought. This solution cannot be determined at random. For all equations, a linear relation is determined that will give the most appropriate solution to the standard structure of the energy equation. Among the many linear relations that can be formed for solution, the most accurate relation as shown in the following equation is taken:Unlike the conditions we present for the solution here, the signs between the body, even, and odd coordinates are opposite. The necessity of working by taking a sign change originating from parity in the odd part in the superspace is seen in our study. When the above solution is made, the following equation is obtained:This equation is called the Euler–Lagrange energy equation in superspace.

Now, we will form the mechanical system with supergraph bundle coordinates. When doing this, we will use the bundle structure, coordinate system, and properties for supergraph bundle that we obtained in Section 2. Here, just differently, we have to work with time because we are studying to obtain the time-dependent Lagrangian energy equation. Actually, the coordinate *u* corresponds to an edge between *x* and *y* vertexes, . On this edge, which we can think of geometrically as a vector from *x* to *y*, the moving particle performs this motion in a time interval *t*. So physically time will be a natural parameter in the 3-dimensional supergraph bundle that we are working with. The coordinates of the supergraph bundle are the same as (19), and only the time relation will be considered. Since only the explication of the edge coordinate is different in bundle coordinates, it will be sufficient to get the geometric structures that form the mechanical system that we have obtained in the first part of Section 3 according to the graph bundle coordinates for the graph bundle because the proofs are similar.

The super almost tangent structure for supergraph bundle is as follows:The semispray with supergraph bundle coordinate is given by