# Euler‐Lagrange Equation in Free Coordinates

Authors: Mastourah M. Alotaibi; Sami H. Altoum
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 classM of function yx and each function y∈M, we define a functional J by J=∫abLx,y,y′dx where y′=dy/dx. Then, J has a stationary condition if the Euler–Lagrange differential equation:(1)∂L∂y−ddx∂L∂y′=0,is satisfied with Lagrangian L=Lx,y,y′. 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 as(2)Fvx=0v=1,…,n,in which F1x,…,Fnx are smooth real valued functions defined for x in some manifold M. 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 Fv must be polynomials—just any differentiable functions. A solution is a point x∈M such that Fvx=0, for v=1,…,n. A symmetry group of the system will be a local group of transformations G acting on M with the property that G transform solutions to other solutions (see [5–8]). In other words, if x is a solution, g is a group element, and g⋅x is defined, then we require that g⋅x is also a solution. G 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 g⋅Definition 1.
LetG be local group of transformations acting on manifold M. A subset φ⊂M is called G-invariant, and G is called symmetry group of φ if whenever x∈φ and g∈G such that g⋅x is defined, then g⋅x∈φ.Definition 2.
Letφ be a system of differential equation. A symmetry group of the system φ is a local group of transformations G acting on an open subset M 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 systemφ of n−order differential equations in p independent and q dependent variables given a system of equations:(3)Δνx,un=0,ν=1,…,s,involving x=x1,…,xp,u=u1,…,uq, and the derivatives of u with respect to x up to order n⋅Δ can be viewed as a smooth map from jet space X×Un for some s−dimensional Euclidean space.Definition 3.
(see [9]). Let G be a local group of transformations acting on a manifold M. A function F:M⟶N, where N is another manifold and is called a G-invariant function if for all x∈M and all g∈G such that g⋅x is defined by(4)Fg⋅x=Fx.
A real-valuedG-invariant function ς:M⟶ℝ is simply called an invariant of G. Such that F:M⟶ℝn, is G -invariant if and only if each components Fn of F=F1,…,Fn is an invariant of G.Definition 4.
LetD⊂M be a bounded, closed set. A variation Ψ in D is a one-parameter family of functions ψs,x, where s∈−ε,ε and x∈M such that(5)Ψ0,x=Ψx,Ψs,x=Ψx;∀x∈M∖D.
Denote(6)δΨxi=∂Ψis,x∂s|s=0,i=1⋯n.Definition 5.
The integral(7)I=∫DLdvg,is called stationary under the above variation if(8)dIdss=0=0.
We denote the volume element(9)dvg=gdx1⋅dx2⋅dx3.Theorem 1.
(see [10]). Equation (7) is stationary under any variation ofΨif and only if the following Euler–Lagrange equations are satisfied:(10)∑k=1m∂L∂Ψ;ki;k=∂L∂Ψi,∀i=1,…,n.Proof.
Applying the chain rule,(11)dIdu∣u=0=∑i∫D∂L∂ΨiδΨi+∂L∂Ψ;eiδΨ;eidvg.
AsδΨi;e=δΨi;e, the second term in the right hand side can be expressed as(12)∑i∫D∂L∂Ψ;eiδΨi;e−∂L∂Ψ;ei;eδΨidvg.
Let(13)X=Xe∂∂xe,where(14)Xe=∑i∂L∂Ψ;eiδΨi,and by the divergence theorem [9],(15)∫DX;eedv=0,as X vanishes on ∂D. Thus,(16)dIds∣s=0=∫D∂L∂Ψi−∂L∂Ψ;ei;eδΨidv=0,for all variotion ofΨ, which means that (14) satisfied. We take the variationΨs,x=expsVΨx, whereVΨx∈TΨxN, and we haveΨ0,x=Ψxand(17)δΨ=∂Ψs,x∂s∣s=0=VΨx,for any arbitrary V.


## 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. LetV be a real vector space whose elements are denoted by v and call “vectors.” A “covector” which is a linear mapping, typically denoted by ω, of V into ℝ, is the real numbers. The space V∗ of covectors forms a new vector space called the dual space to V (see [11–14]).


### 3.1. Dual Space

IfV is finite dimensional, V∗ has the same dimension as V. In fact, suppose v1,…,vn is a basis for V, that is, every element v∈V can be written in the unique form:(18)v=a1v1+⋯+anvn.The coefficientsa1,…,an in this expansion depend linearly on v and hence define linear forms on V, that is, elements of V, which we denote by ω1,…,ωn. That ω1,…,ωn forms a basis for V called the dual basis to the given basis v1,…,vn of V. It can also be characterized by the condition as follows:(19)ωivj=δij,1≤i,j≤n.Anr-covector on V is a mapping(20)v1,…,vr⟶ωv1,…,vr,with domain the n-tuples of elements of V with values in the real numbers. We can indicate this by the notation ω:Vx⋯xV⟶ℝ.We require thatω be multilinear in the sense that it is linear in each of the variables v1,…,vr when all others are held fixed. In addition, we require that it be skew-symmetric in the sense that ωv1,…,vr changes sign when neighboring arguments are permuted, that is,(21)ωv1,…,vr=−ωv2,v1,v3,…,vr,ωv1,v2,v3,…,vr=−ωv1,v3,v2,v4,…,vr.


### 3.2. Wedge Product

A product form denoted byω1∧ω2 can be defined as an r+s-covector. It is called the exterior product of ω1 and ω2. It is obtained in the following way.Considerr+s-vectors v1,…,vr+s. One can assign to them the number(22)ω1v1,…,vrω2vr+1,…,vr+s.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 r=s=1,(23)ω1∧ω2v1,v2=ω1v1ω2v2−ω1v2ω2v1.Notice that from this formula,(24)v⌟ω1∧ω2=+ω1vω2−ω2vω1=v⌟ω1∧ω2−ω1∧v⌟ω2.Ifc is a constant, c∧ω is taken to be just cω. We shall define v⌟ω, the contraction of ω by v, or inner product of ω with v.The general formula can be guessed as follows:(25)v⌟ω1∧ω2=v⌟ω1∧ω2+−1rω1∧v⌟ω2.


## 3.1. Dual Space

IfV is finite dimensional, V∗ has the same dimension as V. In fact, suppose v1,…,vn is a basis for V, that is, every element v∈V can be written in the unique form:(18)v=a1v1+⋯+anvn.The coefficientsa1,…,an in this expansion depend linearly on v and hence define linear forms on V, that is, elements of V, which we denote by ω1,…,ωn. That ω1,…,ωn forms a basis for V called the dual basis to the given basis v1,…,vn of V. It can also be characterized by the condition as follows:(19)ωivj=δij,1≤i,j≤n.Anr-covector on V is a mapping(20)v1,…,vr⟶ωv1,…,vr,with domain the n-tuples of elements of V with values in the real numbers. We can indicate this by the notation ω:Vx⋯xV⟶ℝ.We require thatω be multilinear in the sense that it is linear in each of the variables v1,…,vr when all others are held fixed. In addition, we require that it be skew-symmetric in the sense that ωv1,…,vr changes sign when neighboring arguments are permuted, that is,(21)ωv1,…,vr=−ωv2,v1,v3,…,vr,ωv1,v2,v3,…,vr=−ωv1,v3,v2,v4,…,vr.


## 3.2. Wedge Product

A product form denoted byω1∧ω2 can be defined as an r+s-covector. It is called the exterior product of ω1 and ω2. It is obtained in the following way.Considerr+s-vectors v1,…,vr+s. One can assign to them the number(22)ω1v1,…,vrω2vr+1,…,vr+s.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 r=s=1,(23)ω1∧ω2v1,v2=ω1v1ω2v2−ω1v2ω2v1.Notice that from this formula,(24)v⌟ω1∧ω2=+ω1vω2−ω2vω1=v⌟ω1∧ω2−ω1∧v⌟ω2.Ifc is a constant, c∧ω is taken to be just cω. We shall define v⌟ω, the contraction of ω by v, or inner product of ω with v.The general formula can be guessed as follows:(25)v⌟ω1∧ω2=v⌟ω1∧ω2+−1rω1∧v⌟ω2.


## 4. Examples to N-Forms

Any1−form θ can be written as follows:(26)θ=f1dx1+f2dx2+f3dx3,where dx1,dx2,dx3 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)
dxi∧dxj=−dxj∧dxi⇒dxi∧dxi=0⋅(ii)
Ifη and w are any 1−forms, then ηw+θ can be written as follows:(27)η∧w+θ=η∧w+η∧θ⋅We gives some examples to illustrate, 1-forms, 2-forms and 3-forms respectively.Example 1.
(28)θ=2x1−x2dx1+x32dx2−x1dx3,η=x22dx1−x3dx2+x12dx3.
Then, we get(29)θ∧η=−x32x1−x2dx1dx2+x122x1−x2dx1dx3+x32x22dx2dx1+x12x32dx2dx3−x1x22dx3dx1+x1x3dx3dx2=−x32x1−x2−x32x22dx1dx2+x122x1−x2+x1x22dx1dx3+x12x32−x1x3dx2dx3.Example 2.
(30)Letθ=x1dx1−x2dx2+x32dx3,η=x2dx1dx2−x3dx1dx3+x2dx2dx3.
Then, we get(31)θ∧η=x1x2dx2dx3−x2x3dx1dx2dx3+x2x32dx1dx2dx3=x1x2−x2x3+x2x32dx1dx2d3⋅Example 3.
Let(32)η=x22−x1x32+x32dx2dx3+x2x3dx1⋅
Then, we get(33)dη=−x32dx1dx2dx3+x2dx1dx2dx3=x2−x32dx1dx2dx3⋅


## 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.
A Lagrangian on M is just a real-valued function on TM×ℝ, which is denoted by(34)Lσ=∫abLσ′t,tdt,where σ⟶Lσ on M, σ:a,b⟶M, and t⟶σ′t∈Mσt.


### 5.1. Cartan 1-FormθL

To deduce the Euler–Lagrange equation in free coordinates we use Cartan 1- differential formθL on fM×ℝ. 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 w1,…,wn of differential 1-form in an open set U of M. Adopt the following range of indices and summation convention.(35)1≤i,j,…,≤n=dimM.n∈ℕ.Since no confusion is likely, letwi also denote the differential forms on the open subset of TM×ℝ lying about U, obtained by pulling back, via the projection map. Of course we must then denote by yi the real-valued functions defined by w, on TM, hence also on TM×ℝ.(36)yiv=wivforv∈TM.At any rate,wi,dyi,dt forms a local basis for differential forms on TM×ℝ suppose; then,(37)dL=Liwi+Ln−idyt+Ltdt.Now,(38)θL=Ln+iwi−Ln+iyi−Ldt,and one immediately sees that this reduces to the 1-form, by specializing to the case where wi=dxi, with xi,…,xn functions on U defining a coordinate system. As an independent check on this fact, let us verify that θL remains unchanged when a different basis wi′ of 1-forms for U is used. Suppose that wi′=Aijwj, then also(39)yi′=Aijyj,and(40)dL=Li′wi′+Ln+i′dyi′+Li′dt≡Ln+i′Aijdy,and hence(41)Ln+j=Ln+i′Aijθ′L=Ln+i′wi′−Ln+i′yi′−Ldt=Ln+i′Aijwj−Ln+i′Aijyj−Ldt=Ln+jwj−Ln+jyj−Ldt=θL,which expresses the invariance of θL. We state another property θL: if t⟶σt,a≤t≤b, is a curve in M, if(42)t⟶σ′t,t=γt,it is extended curve on TM×ℝ and then(43)∫abLσ′t,tdt=∫abθLy′tdt⋅Thus, by lifting to the spaceTM×ℝ sitting over M, we have converted a” nonlinear” Lagrangian L to a” linear” one θL. 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 curvet⟶σtonMbe an extremal forL, the Euler–Lagrange equation(44)ddt∂L∂xi−∂L∂xi=0,is equivalent to(45)γ′t⌟dθL=0,if and only if its extended curve t⟶σ′t,t=γt on TM×ℝ is a characteristic curves for the differential form dθL.Proof.
The study of the characteristic curves of a closed2−form is more or less identical with Hamilton—Jacobi theory. Let us work out these conditions explicitly (45) and we find(46)dθL=dLn+iwi−yidt+Ldt=dLn+i∧wi−yidt+Ln+idwi−dyi∧dt+Liwi∧dt+Ln+idyi∧dt=dLn+i∧wi−yidt+Ln+idwi+Liwi∧dt=dLn+i∧wi−yidt+Liwi∧dt+Ln+icjkiwj∧wk,where cjki are the functions such that(47)dwi=cjkiwj∧wk,cjkt+ckji=0,and now(48)wi−yidtγ′t=wiσ′t−yiγt=0dLn+iγ′t=ddtLn+iσ′t,t.
Hence, the conditionsγt=σ′t,t be a characteristic curve of dθL.(49)ddtLn+iσ′t,twi−yidt−Liσ′t,twi+Liyidt+Ln+icjkiyjwk=0.
Therefore, we getwi and dt as independent differential forms; hence, these equations imply that the coefficient of wi is zero, that is,(50)ddtLn+iσ′t,t−Liσ′t,t+Ln+kcjkiσ′t,twjσ′t=0,and these D.Es for σ are the Euler–Lagrange equations with respect to the basis w1,…,wn. Notice in case that w1,…,wn are the differentials of a set of coordinate functions x1,…,xn for M, that they reduce (since cjki=0) to the classical Euler–Lagrange equations:(51)ddt∂L∂x˙i−∂L∂xi=0⋅
If time-derivative notation is replaced instead by space-derivative notation, the equation becomes(52)∂L∂y−ddx∂L∂y′=0.
The 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.


## 5.1. Cartan 1-FormθL

To deduce the Euler–Lagrange equation in free coordinates we use Cartan 1- differential formθL on fM×ℝ. 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 w1,…,wn of differential 1-form in an open set U of M. Adopt the following range of indices and summation convention.(35)1≤i,j,…,≤n=dimM.n∈ℕ.Since no confusion is likely, letwi also denote the differential forms on the open subset of TM×ℝ lying about U, obtained by pulling back, via the projection map. Of course we must then denote by yi the real-valued functions defined by w, on TM, hence also on TM×ℝ.(36)yiv=wivforv∈TM.At any rate,wi,dyi,dt forms a local basis for differential forms on TM×ℝ suppose; then,(37)dL=Liwi+Ln−idyt+Ltdt.Now,(38)θL=Ln+iwi−Ln+iyi−Ldt,and one immediately sees that this reduces to the 1-form, by specializing to the case where wi=dxi, with xi,…,xn functions on U defining a coordinate system. As an independent check on this fact, let us verify that θL remains unchanged when a different basis wi′ of 1-forms for U is used. Suppose that wi′=Aijwj, then also(39)yi′=Aijyj,and(40)dL=Li′wi′+Ln+i′dyi′+Li′dt≡Ln+i′Aijdy,and hence(41)Ln+j=Ln+i′Aijθ′L=Ln+i′wi′−Ln+i′yi′−Ldt=Ln+i′Aijwj−Ln+i′Aijyj−Ldt=Ln+jwj−Ln+jyj−Ldt=θL,which expresses the invariance of θL. We state another property θL: if t⟶σt,a≤t≤b, is a curve in M, if(42)t⟶σ′t,t=γt,it is extended curve on TM×ℝ and then(43)∫abLσ′t,tdt=∫abθLy′tdt⋅Thus, by lifting to the spaceTM×ℝ sitting over M, we have converted a” nonlinear” Lagrangian L to a” linear” one θL. 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 curvet⟶σtonMbe an extremal forL, the Euler–Lagrange equation(44)ddt∂L∂xi−∂L∂xi=0,is equivalent to(45)γ′t⌟dθL=0,if and only if its extended curve t⟶σ′t,t=γt on TM×ℝ is a characteristic curves for the differential form dθL.Proof.
The study of the characteristic curves of a closed2−form is more or less identical with Hamilton—Jacobi theory. Let us work out these conditions explicitly (45) and we find(46)dθL=dLn+iwi−yidt+Ldt=dLn+i∧wi−yidt+Ln+idwi−dyi∧dt+Liwi∧dt+Ln+idyi∧dt=dLn+i∧wi−yidt+Ln+idwi+Liwi∧dt=dLn+i∧wi−yidt+Liwi∧dt+Ln+icjkiwj∧wk,where cjki are the functions such that(47)dwi=cjkiwj∧wk,cjkt+ckji=0,and now(48)wi−yidtγ′t=wiσ′t−yiγt=0dLn+iγ′t=ddtLn+iσ′t,t.
Hence, the conditionsγt=σ′t,t be a characteristic curve of dθL.(49)ddtLn+iσ′t,twi−yidt−Liσ′t,twi+Liyidt+Ln+icjkiyjwk=0.
Therefore, we getwi and dt as independent differential forms; hence, these equations imply that the coefficient of wi is zero, that is,(50)ddtLn+iσ′t,t−Liσ′t,t+Ln+kcjkiσ′t,twjσ′t=0,and these D.Es for σ are the Euler–Lagrange equations with respect to the basis w1,…,wn. Notice in case that w1,…,wn are the differentials of a set of coordinate functions x1,…,xn for M, that they reduce (since cjki=0) to the classical Euler–Lagrange equations:(51)ddt∂L∂x˙i−∂L∂xi=0⋅
If time-derivative notation is replaced instead by space-derivative notation, the equation becomes(52)∂L∂y−ddx∂L∂y′=0.
The 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.



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