Euler‐Lagrange Equation in Free Coordinates

Abstract

In this paper, we introduce different equivalent formulations of variational principle. The language of differential forms and manifold has been utilized to deduce Euler–Lagrange equations in free coordinates. Thus, the expression is simple and global.

1. Introduction

The calculus of variation is one of the most important divisions of classical mathematical analysis with regard to application. Basic machinery of calculus of variation is introducing variations, which are small changes in functions and functionals to find maxima and minima of functionals, which is a useful way of developing the differential equations for extremal in terms of arbitrary basis of differential forms. The Euler–Lagrange is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. If we have a certain class of function and each function , we define a functional by where . Then, has a stationary condition if the Euler–Lagrange differential equation:is satisfied with Lagrangian . The application of manifold theory to geometry involves the study of properties of volumes, curvature, and the system of ordinary differential equations. In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus, that is, independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds (see [1–4]). The great advantages in certain types of calculation are, namely, the use of exterior calculus characterized by differential forms to deduce an expression of Euler–Lagrange equations. The expression will be simple, global, and independent of coordinates. This manuscript arranging is as follows. In section 2, we introduce the notions of symmetry groups on manifold. In section 3, we present differential forms. In section 4, we introduce some examples to n-forms. In section 5, we deduce the Euler–Lagrange equation in free coordinates.

2. Preliminaries

Before considering symmetry group of differential equations, it is essential that we deal properly with the conceptually simpler case of symmetry groups of systems of algebraic equations by a” system of algebraic equations”; we mean a system of equations asin which are smooth real valued functions defined for in some manifold . Note that the adjective “algebraic” is only used to distinguish this case from the case of systems of differential equations; it does not mean that must be polynomials—just any differentiable functions. A solution is a point such that , for . A symmetry group of the system will be a local group of transformations acting on with the property that transform solutions to other solutions (see [5–8]). In other words, if is a solution, is a group element, and is defined, then we require that is also a solution. is a symmetry group, the elements of this group are functions, and the property of this group takes solution to another solution with the preserving the structure. This group has at least one element called identity, which denoted by the symbol

Definition 1. Let be local group of transformations acting on manifold . A subset is called G-invariant, and is called symmetry group of if whenever and such that is defined, then .

Definition 2. Let be a system of differential equation. A symmetry group of the system is a local group of transformations acting on an open subset of the space of independent and dependent variables for the system with the property that whenever is also a solution of the system.
The system of D.E. is a system of order differential equations in independent and dependent variables given a system of equations:involving , and the derivatives of with respect to up to order can be viewed as a smooth map from jet space for some dimensional Euclidean space.

Definition 3. (see [9]). Let be a local group of transformations acting on a manifold . A function , where is another manifold and is called a -invariant function if for all and all such that is defined byA real-valued -invariant function is simply called an invariant of . Such that , is -invariant if and only if each components of is an invariant of .

Definition 4. Let be a bounded, closed set. A variation in is a one-parameter family of functions , where and such thatDenote

Definition 5. The integralis called stationary under the above variation ifWe denote the volume element

Theorem 1. (see [10]). Equation (7) is stationary under any variation ofif and only if the following Euler–Lagrange equations are satisfied:

Proof. Applying the chain rule,As , the second term in the right hand side can be expressed asLetwhereand by the divergence theorem [9],as vanishes on . Thus,for all variotion of, which means that (14) satisfied. We take the variation, where, and we haveandfor any arbitrary .

3. Differential Forms

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariables, calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. Let be a real vector space whose elements are denoted by and call “vectors.” A “covector” which is a linear mapping, typically denoted by , of into , is the real numbers. The space of covectors forms a new vector space called the dual space to (see [11–14]).

3.1. Dual Space

If is finite dimensional, has the same dimension as . In fact, suppose is a basis for , that is, every element can be written in the unique form:

The coefficients in this expansion depend linearly on and hence define linear forms on , that is, elements of , which we denote by . That forms a basis for called the dual basis to the given basis of . It can also be characterized by the condition as follows:

An -covector on is a mappingwith domain the -tuples of elements of with values in the real numbers. We can indicate this by the notation .

We require that be multilinear in the sense that it is linear in each of the variables when all others are held fixed. In addition, we require that it be skew-symmetric in the sense that changes sign when neighboring arguments are permuted, that is,

3.2. 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 (see [15–18]). It can be made so by permuting the variables and adding up the results, with appropriate signs. For example, if ,

Notice that from this formula,

If is a constant, is taken to be just We shall define , the contraction of by , or inner product of with The general formula can be guessed as follows:

4. Examples to N-Forms

Any form can be written as follows:where is regarded as a basis for the space of all.

1—forms, 2—forms, and 3—forms:

We need to define a product (called wedge product) in the following way:(i)(ii)If and are any forms, then can be written as follows:We gives some examples to illustrate, 1-forms, 2-forms and 3-forms respectively.

Example 1. Then, we get

Example 2. Then, we get

Example 3. LetThen, we get

5. Derivation of Variational Principle Using Differential Forms

In this section, we shall pause to show that the basic machinery of calculus of variation can be formulated in coordinate free. As a bouns, we shall obtain a useful way of developing the differential equation of extremals in terms of an arbitrary basis of differential forms.

Definition 6. Lagrangian on is just a real-valued function on , which is denoted bywhere on , , and .

5.1. Cartan 1-Form

To deduce the Euler–Lagrange equation in free coordinates we use Cartan 1- differential form on . This can be defined by using a coordinate system. We find it useful to see a more general definition in terms of an arbitrary basis of differential 1-form in an open set of . Adopt the following range of indices and summation convention.

Since no confusion is likely, let also denote the differential forms on the open subset of lying about , obtained by pulling back, via the projection map. Of course we must then denote by the real-valued functions defined by , on , hence also on .

At any rate, forms a local basis for differential forms on suppose; then,

Now,and one immediately sees that this reduces to the 1-form, by specializing to the case where , with functions on defining a coordinate system. As an independent check on this fact, let us verify that remains unchanged when a different basis of 1-forms for is used. Suppose that , then alsoandand hencewhich expresses the invariance of . We state another property : if , is a curve in , ifit is extended curve on and then

Thus, by lifting to the space sitting over , we have converted a” nonlinear” Lagrangian to a” linear” one . This procedure of lifting to a higher-dimensional space to simplify the structure of a geometric object is typical of Cartan’s entire approach to differential geometry. From above discussion, we introduce the following theorem.

Theorem 2. Let a curveonbe an extremal for, the Euler–Lagrange equationis equivalent toif and only if its extended curve on is a characteristic curves for the differential form .

Proof. The study of the characteristic curves of a closed form is more or less identical with Hamilton—Jacobi theory. Let us work out these conditions explicitly (45) and we findwhere are the functions such thatand nowHence, the conditions be a characteristic curve of .Therefore, we get and as independent differential forms; hence, these equations imply that the coefficient of is zero, that is,and these s for are the Euler–Lagrange equations with respect to the basis . Notice in case that are the differentials of a set of coordinate functions for , that they reduce (since ) to the classical Euler–Lagrange equations:If time-derivative notation is replaced instead by space-derivative notation, the equation becomesThe more general equations are very useful in certain mechanical problems (for example, in rigid body dynamics) where a basis for differential forms can be found more readily than for the natural coordinate system.

6. Conclusion

In this paper, we utilized methods of global analysis such as differential geometry. We deduce Euler–Lagrange equations, and the new expression is global and free of coordinates. The old formulae are constrained by coordinates.

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant code: 22UQU4310382DSR02, funded by authors personal budget.