This work presents the geometrical formulation of variational principle and uses exterior calculus characterized by differential forms, smooth manifolds, and the theory of fiber bundle to deduce variational principle in particle field, and we show the difference between classical form and new form.

1. Introduction

Many problems in Newtonian mechanics are more easily analyzed by means of alternative statements of laws, including Hamiltonian principle and Lagrange’s equation, and these equations are derived from some general principles of calculus of variations. Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Mathematically, this involves finding stationary values of integrals of the form.


Equation (1) is called the Euler–Lagrange equation, and the solution of equation (1) is the min or max curve. The Euler–Lagrange equation appears in many scientific applications, especially in physics. The fiber bundle theory is an important part of pure mathematics, in particular in the field of differential geometry. A principle fiber bundle is a special kind of fiber bundle which is a bundle whose fiber is a Lie group. Principle bundles have important applications in differential geometry, topology, and physics, especially in gauge theory[14]. In our study, we formulated the variation by modern methods of differential geometry via differential forms, that lead to global variation of fiber bundles, which satisfies Euler Lagrange equation . The application of manifold theory to geometry involves the study of properties of volumes, curvature, and the system of ordinary differential equations. A principle fiber bundle (PFB) is a fiber bundle in which fiber is equal to structure group to itself where : such that the following two conditions hold:(i), : is a diffeomorphism and . The multiplication : , defined by , is a diffeomorphism.(ii) is one to one map and [58]. Here denotes the fiber above .

We formulate the principle of least action for particle fields under the influence of gauge potential. In this study, we introduce the geometrical formulation of (VP), using the theory of differential forms, smooth manifolds, and the theory of fiber bundle, to get Euler–Lagrange equation in particle field, and we show the difference between classical form and new form. This study is arranged as follows. In Section 1, we present variational principle and the notions of fiber bundle. In Section 2, we introduce vector bundles and differential forms. In Section 3, we explain particle field and gauge invariance. In Section 4, we derive variational principle (VP) using fiber bundle.

2. Vector Bundles and Differential Forms

Definition 1. Let be the Lie algebra of . A connection is a -valued 1-form defined on such that properties (a) and (b) hold:(a)Let and be the vector field on defined by. is called fundamental field.(b)For , let . We require for all , and . In other words, , when is a connection 1-form [912].

2.1. Wedge Product

A product form denoted by can be defined as an -covector. It is called the exterior product of and . It is obtained in the following way.

Consider -vectors . One can assign to them the number

However, this assignment does not depend skew-symmetrically on all variables [1316]. It can be made so by permuting the variables and adding up the results, with appropriate signs. For example, if ,

From this formula, it follows that

If is a constant, is taken to be just . We define by inner product. The general formula can be guessed as

2.2. Examples to N-Forms

Any form can be written as

The exterior product of 1-form at any point satisfy the properties of skew symmetric. Also, the exterior product satisfy the properties ofdistributivity bilinear, associativity, and anticommutative [1316]. We can define 1-form as follows:where are basis.

The wedge product is defined as follows. (1) . (2) Let and be any forms; then, we get

Example 1. LetThen, we get

Example 2. Then, we get

Example 3. LetThen, we get

3. Particle Fields and Gauge Invariance

Let : be a PFB with . Let act on manifold . , define the map when ) where , and , therefore, is . If is linear, then the homomorphism given by is called a representation of where .

Let be the space of map where . Suppose a connection on . Also note that is equivalent to ([17, 18]).

3.1. Gauge Invariance and Lagrangians

Let be a PFB with group and be representation. The space of 1-jets of maps from to is

A Lagrangian is a map : , where and ; then,

As result of this requirement, we have the following theorem.

Theorem 1. The Lagrangian : is definedby a function , when , , and as by,

Proof. Show that is independent of . Since , then we haveorTherefore, we get

Theorem 2 (see [7]). Let : be a -invariant Lagrangian, and let be the space of connection on . Define a functionbyfor , , , and ; then, is not only well defined but also gauge invariant in the sense that for , we get

Definition 2. For , we define the projected support of to be closure of set ; then we say that is stationary relative to if , and also with projected , support contained in , we have,Equivalently, we say that obeys the principle of least action. The remainder is devoted to showing that is stationary if satisfies a certain differential equation (Lagrange’s equation). This amounts to mimicking the usual approach to calculus of variation problems, but the general setting here forces some additional notations on us.

4. Derivation of Variational Principle Using Fiber Bundle

Let be a Lagrangian and let indicate the space of linear maps . For , define by the equation

For , define a -valued 1-form on by

Also, for , define by

For , define a -valued 1-form on by

Theorem 3. From equation (29), we have that and .

Proof. Note that disappear on vertical vectors by definition. Then, we getNow, let . Then, we getThus,Similarly, we prove .
From Theorem 3, we deduce the following theorem.

Theorem 4. Suppose that and let , with projected support on ; then, at , we get

Proof. At , we getIntegrating both sides over , applying Theorem 3, and noting that on , we get the result.

Theorem 5. The particle field is stationary for a Lagrangian and a fixed connection on if and only if the Lagrange equation holds:

Proof. Assume that the Euler–Lagrange equation does not hold at , so we find ; therefore,By continuity, this inequality persists in an open set containing . Multiplying by a positive function with projected support in and with , we obtain . Theorem 4 applies to , and the R.H.S of the equation in Theorem 4 is nonzero. Thus,and is not stationary. The converse is clear from Theorem 4.

5. Conclusion

In this work, we use advance calculus and differential forms in order to deduce variational principle in particle field. (1) is the Euler–Lagrange equation in classical case, and (35) is the Euler–Lagrange equation in particle field; the new formula is simple and comprehensive in particle fields. We expect to study the nanoparticle and quantum white noise [1923] case which is now attractive in mathematical physics.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work (grant code: 22UQU4310382DSR05).