# The Geometrical Formulation of Variational Principle under the Theory of Fiber Bundle

Authors: Reem A. Alrebdi; Sami H. Altoum
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 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.Then,(1)∂f∂y−ddx∂f∂y′=0.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[1–4]. 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 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. A principle fiber bundle (PFB) is a fiber bundle in which fiber F is equal to structure group G to itself where Π: P⟶M such that the following two conditions hold:(i)
∀g∈G, Rg: P⟶P is a diffeomorphism and Rgp=pg. The multiplication Rg: P⟶P, defined by Rgp=pg, is a diffeomorphism.(ii)
Π:P⟶M is one to one map and Π−1Πp=pg:g [5–8]. Here Π−1x denotes the fiber above x.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 Section1, 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.
Letϑ be the Lie algebra of G. A connection is a ϑ-valued 1-form ω defined on p such that properties (a) and (b) hold:(a)
LetA∈ϑ and A∗ be the vector field on P defined by(2)Ap∗=ddtpexptA|t=0.Ap∗=A. Ap∗ is called fundamental field.(b)
Forg∈G, let Gg:ϑ⟶ϑ. We require ωpgRgX=Gg−1ωpX for all g∈G, p∈P and X∈TpP. In other words, Rg∗ω=Gg−1ω, when ω is a connection 1-form [9–12].


### 2.1. 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(3)ω1v1,…,vrω2vr+1,…,vr+s.However, this assignment does not depend skew-symmetrically on all variables [13–16]. It can be made so by permuting the variables and adding up the results, with appropriate signs. For example, if r=s=1,(4)ω1∧ω2v1,v2=ω1v1ω2v2−ω1v2ω2v1.From this formula, it follows that(5)v⌟ω1∧ω2=+ω1vω2−ω2vω1=v⌟ω1∧ω2−ω1∧v⌟ω2⋅Ifc is a constant, c∧ω is taken to be just cω. We define by inner product. The general formula can be guessed as(6)v⌟ω1∧ω2=v⌟ω1∧ω2+−1rω1∧v⌟ω2.


### 2.2. Examples to N-Forms

Any1−form θ can be written as(7)θ=f1dx1+f2dx2+f3dx3.The exterior product of 1-form at any pointp satisfy the properties of skew symmetric. Also, the exterior product satisfy the properties ofdistributivity bilinear, associativity, and anticommutative [13–16]. We can define 1-form as follows:(8)ξ=f1dx1+f2dx2+f3dx3,where dx1,dx2,dx3 are basis.The wedge product is defined as follows. (1)dxi∧dxj=−dxj∧dxi⇒dxi∧dxi=0. (2) Let ζ and γ be any 1−forms; then, we get(9)ζ∧γ+ξ=ζ∧γ+ζ∧ξ.Example 1.
Let(10)θ=2x1−x2dx1+x32dx2−x1dx3η=x22dx1−x3dx2+x12dx3.
Then, we get(11)θ∧η=−x32x1−x2dx1dx2+x122x1−x2dx1dx3+x32x22dx2dx1+x12x32dx2dx3−x1x22dx3dx1+x1x3dx3dx2,=−x32x1−x2−x32x22dx1dx2+x122x1−x2+x1x22dx1dx3+x12x32−x1x3dx2dx3⋅Example 2.
(12)Letθ=x1dx1−x2dx2+x32dx3,η=x2dx1dx2−x3dx1dx3+x2dx2dx3.
Then, we get(13)θ∧η=x1x2dx2dx3−x2x3dx1dx2dx3+x2x32dx1dx2dx3=x1x2−x2x3+x2x32dx1dx2d3.Example 3.
Let(14)η=x22−x1x32+x32dx2dx3+x2x3dx1.
Then, we get(15)dη=−x32dx1dx2dx3+x2dx1dx2dx3=x2−x32dx1dx2dx3.


## 2.1. 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(3)ω1v1,…,vrω2vr+1,…,vr+s.However, this assignment does not depend skew-symmetrically on all variables [13–16]. It can be made so by permuting the variables and adding up the results, with appropriate signs. For example, if r=s=1,(4)ω1∧ω2v1,v2=ω1v1ω2v2−ω1v2ω2v1.From this formula, it follows that(5)v⌟ω1∧ω2=+ω1vω2−ω2vω1=v⌟ω1∧ω2−ω1∧v⌟ω2⋅Ifc is a constant, c∧ω is taken to be just cω. We define by inner product. The general formula can be guessed as(6)v⌟ω1∧ω2=v⌟ω1∧ω2+−1rω1∧v⌟ω2.


## 2.2. Examples to N-Forms

Any1−form θ can be written as(7)θ=f1dx1+f2dx2+f3dx3.The exterior product of 1-form at any pointp satisfy the properties of skew symmetric. Also, the exterior product satisfy the properties ofdistributivity bilinear, associativity, and anticommutative [13–16]. We can define 1-form as follows:(8)ξ=f1dx1+f2dx2+f3dx3,where dx1,dx2,dx3 are basis.The wedge product is defined as follows. (1)dxi∧dxj=−dxj∧dxi⇒dxi∧dxi=0. (2) Let ζ and γ be any 1−forms; then, we get(9)ζ∧γ+ξ=ζ∧γ+ζ∧ξ.Example 1.
Let(10)θ=2x1−x2dx1+x32dx2−x1dx3η=x22dx1−x3dx2+x12dx3.
Then, we get(11)θ∧η=−x32x1−x2dx1dx2+x122x1−x2dx1dx3+x32x22dx2dx1+x12x32dx2dx3−x1x22dx3dx1+x1x3dx3dx2,=−x32x1−x2−x32x22dx1dx2+x122x1−x2+x1x22dx1dx3+x12x32−x1x3dx2dx3⋅Example 2.
(12)Letθ=x1dx1−x2dx2+x32dx3,η=x2dx1dx2−x3dx1dx3+x2dx2dx3.
Then, we get(13)θ∧η=x1x2dx2dx3−x2x3dx1dx2dx3+x2x32dx1dx2dx3=x1x2−x2x3+x2x32dx1dx2d3.Example 3.
Let(14)η=x22−x1x32+x32dx2dx3+x2x3dx1.
Then, we get(15)dη=−x32dx1dx2dx3+x2dx1dx2dx3=x2−x32dx1dx2dx3.


## 3. Particle Fields and Gauge Invariance

LetΠ: P⟶M be a PFB with G. Let G act on manifold F. ∀g∈G, define the mapLg:F⟶F when g⋅f=Lgf) where e⋅f=f, g1g2⋅f=g1⋅g2⋅f and G×F⟶F, therefore, g,f⟶g⋅f is C∞. If Lg:V⟶V is linear, then the homomorphism G⟶GLV given by g⟶Lg is called a representation of G where G⟶GLV.LetCP,F be the space of map τ:P⟶F where τpg=g−1⋅τp. Suppose a connection ω on PΩω∈∧¯2P,ϑ. Also note that CP,V is equivalent to ∧¯0P,V ([17, 18]).


### 3.1. Gauge Invariance and Lagrangians

LetΠ:P⟶M be a PFB with group G and G⟶GLV be representation. The space of 1-jets of maps from P to V is(16)JP,V=p,v,θ,p∈P,v∈Vandθ:TpP⟶Vislinear.A Lagrangian is a mapL: JP,V⟶ℝ, where p,v,θ∈JP,V and g∈G; then,(17)Lpg,g−1⋅v,g−1⋅θ°Rg−1∗=Lp,v,θ.As result of this requirement, we have the following theorem.Theorem 1.
The LagrangianL: JP,V⟶ℝ is definedby a function £0:CP,V⟶C∞M, when x∈M, ψ∈CP,V, and p∈P as Πp=xby,(18)E0ψx=Lp,ψp,dψp.Proof.
Show thatLp,ψp,dψp is independent of p∈Π−1x. Since ψ∘Rg=g−1⋅ψ, then we have(19)dψpg∘Rg∗=g−1⋅dψp,or(20)dψpg=g−1⋅dψp∘Rg−1∗.
Therefore, we get(21)Lpg,ψpg,dψpg=Lpg,g−1⋅ψp,g−1⋅dψpRg−1∗=Lp,ψp,dψp.Theorem 2 (see [7]).
LetL: JP,V⟶ℝ be a G-invariant Lagrangian, and let ℓ be the space of connection on P. Define a function(22)£:CP,V×ℓ⟶C∞M,by(23)£ψ,ωx=Lp,ψp,Dωψp,for x∈M, p∈Π−1x, ψ∈CP,V, and ω∈ℓ; then, E is not only well defined but also gauge invariant in the sense that for f∈GAP, we get(24)£f∗ψ,f∗ω=£ψ,ω.Definition 2.
Forψ∈CP,V, we define the projected support of ψ to be closure of set Πp∈M∣ψp≠0; then we say that ψ∈CP,Vis stationary relative to £ω if ∀U⊂M, and also with projected σ∈CP,V, support contained in U, we have,(25)ddt£¯Uωψ+tσ∣t=0=0.
Equivalently, we say thatψ obeys the principle of least action. The remainder is devoted to showing that ψ∈CP,V is stationary if ψ satisfies a certain differential equation (Lagrange’s equation). This amounts ψ to mimicking the usual approach to calculus of variation problems, but the general setting here forces some additional notations on us.


## 3.1. Gauge Invariance and Lagrangians

LetΠ:P⟶M be a PFB with group G and G⟶GLV be representation. The space of 1-jets of maps from P to V is(16)JP,V=p,v,θ,p∈P,v∈Vandθ:TpP⟶Vislinear.A Lagrangian is a mapL: JP,V⟶ℝ, where p,v,θ∈JP,V and g∈G; then,(17)Lpg,g−1⋅v,g−1⋅θ°Rg−1∗=Lp,v,θ.As result of this requirement, we have the following theorem.Theorem 1.
The LagrangianL: JP,V⟶ℝ is definedby a function £0:CP,V⟶C∞M, when x∈M, ψ∈CP,V, and p∈P as Πp=xby,(18)E0ψx=Lp,ψp,dψp.Proof.
Show thatLp,ψp,dψp is independent of p∈Π−1x. Since ψ∘Rg=g−1⋅ψ, then we have(19)dψpg∘Rg∗=g−1⋅dψp,or(20)dψpg=g−1⋅dψp∘Rg−1∗.
Therefore, we get(21)Lpg,ψpg,dψpg=Lpg,g−1⋅ψp,g−1⋅dψpRg−1∗=Lp,ψp,dψp.Theorem 2 (see [7]).
LetL: JP,V⟶ℝ be a G-invariant Lagrangian, and let ℓ be the space of connection on P. Define a function(22)£:CP,V×ℓ⟶C∞M,by(23)£ψ,ωx=Lp,ψp,Dωψp,for x∈M, p∈Π−1x, ψ∈CP,V, and ω∈ℓ; then, E is not only well defined but also gauge invariant in the sense that for f∈GAP, we get(24)£f∗ψ,f∗ω=£ψ,ω.Definition 2.
Forψ∈CP,V, we define the projected support of ψ to be closure of set Πp∈M∣ψp≠0; then we say that ψ∈CP,Vis stationary relative to £ω if ∀U⊂M, and also with projected σ∈CP,V, support contained in U, we have,(25)ddt£¯Uωψ+tσ∣t=0=0.
Equivalently, we say thatψ obeys the principle of least action. The remainder is devoted to showing that ψ∈CP,V is stationary if ψ satisfies a certain differential equation (Lagrange’s equation). This amounts ψ 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

LetL:JP,V⟶ℝ be a Lagrangian and let ∧¯1P,Vp indicate the space of linear maps TpP⟶V. For p,v,θ∈JP,V, define ∇3Lp,v,θ∈∧¯P,Vp by the equation(26)h¯h^∇3Lp,v,θ,β=ddtLp,v,θ+tβ∣t=0.Forψ∈CP,V, define a V-valued 1-form ∂L/∂Dωψ on P by(27)∂L∂Dωψ=∇3Lp,ψp,Dωψp.Also, forp,v,θ∈JP,V, define ∇2Lp,v,θ∈V by(28)h^∇2Lp,v,θ,w=ddtLp,v+tw,θ∣t=0.Forψ∈CP,V, define a V-valued 1-form ∂L/∂ψ on P by(29)∂L∂ψ=∇2Lp,ψpr,Dωψp.Theorem 3.
From equation (29), we have that ∂L/∂ψ∈CP,V and ∂L/∂Dωψ∈∧¯1P,Vp.Proof.
Note that∂L/∂Dωψ disappear on vertical vectors by definition. Then, we get(30)Rg∗∂L∂Dωψ=g−1∂L∂Dωψ.
Now, letβ∈∧¯1P,V. Then, we get(31)h¯h^pg⋅∂L∂Dωψpg∘Rg∗,βp,=h¯h^pg∂L∂Dωψpg,g−1βp∘Rg−1∗,=h¯h^pg∂L∂Dωψpg,βpg,=ddtLpg,ψpg,Dωψpg+tβpg∣t=0,=ddtLpg,g−1.ψp,Dωψp+tβp∘Rg−1∗∣t=0,=ddtLp,ψp,Dωψp+tβp∣t=0=h¯h^p∂L∂Dωψp,βp.
Thus,(32)g.∂L∂Dωψpg∘Rg∗=∂L∂Dωψp∈∧¯1P,V.
Similarly, we prove∂L/∂ψ∈CP,V.
From Theorem3, we deduce the following theorem.Theorem 4.
Suppose thatU⊂M and let τ∈CP,V, with projected support on U; then, at t=0, we get(33)ddt∫U£ωψ+tτμ=∫Uh^δω∂L/∂Dωψ+∂L/∂ψ,τμ.Proof.
Att=0, we get(34)ddt£ωψ+tτΠp=ddtLp,ψp+tτp,Dωψp+tDωτp=h^∂L∂ψp,τp+h¯h^∂L∂Dωψp,Dωτp⋅
Integrating both sides overU, applying Theorem 3, and noting that h¯h^=h^ on ∧¯0P,V=CP,V, we get the result.Theorem 5.
The particle fieldψ∈CP,V is stationary for a Lagrangian L:JP,V⟶ℝ and a fixed connection ω on P if and only if the Lagrange equation holds:(35)δω∂L∂Dωψ+∂L∂ψ=0.Proof.
Assume that the Euler–Lagrange equation does not hold atp∈P, so we find τ¯∈CP,V; therefore,(36)h^δω∂L∂Dωψ+∂L∂ψ,τ¯>0,atΠp.
By continuity, this inequality persists in an open setU containing Πp. Multiplying τ¯ by a positive C∞ function f:P⟶ℝ with projected support in U and with fp〉0, we obtain τ=fτ¯∈CP,V. Theorem 4 applies to τ, and the R.H.S of the equation in Theorem 4 is nonzero. Thus,(37)ddt∫U£ωψ+tτμ∣t=0≠0,and ψ 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 [19–23] case which is now attractive in mathematical physics.



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