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# Approximate symmetries of geodesic equations on 2-spheres

K. Saifullah  and K. Usman
Department of Mathematics
Institute of Applied Mathematics
Technische Universität    Dortmund    Germany

###### Abstract

Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmetries of the approximate geodesic equations show that no non-trivial approximate symmetry for these spaces exists.

preprint: \dedicated

## 1 Introduction

For nonlinear problems, analytic solutions are rare and hard to obtain. Lie group theory provide a systematic and unified approach in search of analytical solutions [1, 2, 3]. Instead of solving the nonlinear equations directly, a system of over determined linear equations is studied using the theory. Once the so-called symmetries of the equations are calculated, similarity solutions can be produced. By defining canonical coordinates, it is also possible to transform the equations into a much simpler form. Another approach allows to find approximate symmetries of approximate differential equations. Generally, the perturbed term in a differential equation corresponds to some small error or correction. Therefore the resulting equations are defined approximately depending on a small parameter and such equations occur frequently in applications. The theory of approximate groups provides a regular method of calculating the perturbation directly, without using the complicated group transformations. The calculation is based on approximate symmetry equations. Recently, approximate symmetries have been used to find the approximate solutions of some partial differential equations [4, 5] as well.

Let us consider an approximate (perturbed) equation in variables and , with a small parameter , i.e.

 E(t,y,ϵ)≈0. (1)

We write it as

 E(t,y,ϵ)≡E0(t,y)+ϵE1(t,y)≈0, (2)

where and are respectively the exact and approximate parts of the approximate differential equation (1). For (2) the infinitesimal generator can be written as [1, 2]

 X=X0+ϵX1, (3)

where

 X0 = ξ0∂∂t+η0∂∂y, X1 = ξ1∂∂t+η1∂∂y.

Here , , and are all functions of and . is the generator of the exact differential equation , and is the generator of the approximate differential equation . Hence we can write

 X=(ξ0+ϵξ1)∂∂t+(η0+ϵη1)∂∂y.

More generally

 X=ξ∂∂t+η∂∂y,

where

 ξ = ξ0+ϵξ1, η = η0+ϵη1.

A second prolongation [1, 2] is given by

 X=ξ∂∂t+η∂∂y+η(1)∂∂y′+η(2)∂∂y′′.

Equation (2) is approximately invariant under the approximate group of transformations with the generator given in (3) if and only if , or

 [X0E0+ϵ(X0E1+X1E0)]∣E≈0=o(ϵ). (4)

This is the determining equation and is called an infinitesimal approximate symmetry or an approximate symmetry or an approximate operator , admitted by (2) .

If is a generator of the unperturbed differential equation

 E0=0, (5)

obtained by solving the determining equation

 X0E0∣E≈0=0, (6)

then we define the auxiliary function by

 H=1ϵX0(E0+ϵE1)∣E0+ϵE1=0, (7)

and find an approximate symmetry, (3), of the perturbed differential equation (2) by solving for  in

 X1E0∣E≈0+H=0. (8)

Note that (8), unlike the determining equation (6) for exact symmetries, is non-homogeneous.

The geodesic equations on a manifold can be regarded as a system of second order ordinary differential equations. The curvature of surfaces can be positive, zero or negative. Symmetries of geodesic equations for these three types of manifolds have been studied . Interesting connection between these symmetries and the isometries  of these spaces have also been found. In this paper we explore approximate symmetries of geodesic equations of 2-spheres. Perturbations in geodesic equations correspond to approximate path of the particle as opposed to the exact one. It may be noted that we study approximate geodesics on a 2-manifold rather than exact geodesics on a perturbed manifold.

## 2 Approximate symmetries of geodesic equations on a 2-sphere

A geodesic is a locally length-minimizing curve and give the shortest distance between two points. Equivalently, it is a path that a particle which is not accelerating would follow. The geodesic equations on a manifold can be regarded as a system of second order ordinary differential equations. In coordinates these can be expressed as [8, 9]

 ¨xa+Γabc˙xb˙xc=0, (9)

where dot represents derivative with respect to the arc length parameter and denote the Christoffel symbols, which for a metric , are defined by

Here denotes the inverse of the metric and the range of the indices gives the dimension of the space.

Exact symmetries of the unit sphere have been given in Ref. . We first outline the derivation of these symmetries and then use them for constructing the approximate symmetries. The metric for the sphere can be written as

 ds2=dθ2+sin2θdϕ2.

The geodesic equations (9) for this metric are given by

 E1 : ¨θ−sinθcosθ˙ϕ2=0, E2 : ¨ϕ+2cotθ˙θ˙ϕ=0. (11)

These are second order ordinary differential equations. We apply the second prolongation ,

 X=ξ∂∂s+η1∂∂θ+η2∂∂ϕ+η1,s∂∂˙θ+η2,s∂∂˙ϕ+η1,ss∂∂¨θ+η2,ss∂∂¨ϕ, (12)

of the symmetry generator given by (3), to both the geodesic equations. Here , and are all functions of , and ; and are functions of , , , and ; and and are all functions of , , , , , and , Then

 XE1 ∣ E1=0=E2=0, XE2 ∣ E1=0=E2=0,

respectively yield 

 [η1,ss−2sinθcosθ˙ϕη2,s−η1(cos2θ−sin2θ)˙ϕ2]∣E1=0=E2=0, (13)
 [η2,ss+2cotθ(˙ϕη1,s+˙θη2,s)−2˙θ˙ϕcsc2θη1]∣E1=0=E2=0. (14)

If we write

 D=∂∂s+˙θ∂∂θ+˙ϕ∂∂ϕ+¨θ∂∂˙θ+¨ϕ∂∂˙ϕ, (15)

then

 η1,s=Dη1−˙θDξ=η1s+˙θ(η1θ−ξs)−˙θ2ξθ+˙ϕη1ϕ−˙θ˙ϕξϕ, (16)
 η2,s=Dη2−˙ϕDξ=η2s+˙ϕ(η2ϕ−ξs)+˙θη2θ−˙ϕ2ξϕ−˙θ˙ϕξθ, (17)
 η1,ss = Dη1,s−¨θDξ (18) = η1ss+˙θ(2η1sθ−ξss)+˙θ2(η1θθ−2ξsθ)−˙θ3ξθθ+2˙ϕη1sϕ +˙ϕ2η1ϕϕ+2˙θ˙ϕ(η1θϕ−ξsϕ)−2˙θ2˙ϕξθϕ−˙θ˙ϕ2ξϕϕ +¨θ(η1θ−2ξs−3˙θξθ)−2¨θ˙ϕξϕ+¨ϕη1ϕ−˙θ¨ϕξϕ,
 η2,ss = Dη2,s−¨ϕDξ (19) = η2ss+2˙θη2sθ+˙θ2η2θθ+˙ϕ(2η2sϕ−ξss)+˙ϕ2(η2ϕϕ−2ξsϕ) −˙ϕ3ξϕϕ+2˙θ˙ϕ(η2θϕ−ξsθ)−˙θ2˙ϕξθθ−2˙θ˙ϕ2ξθϕ+¨θη2θ −¨θ˙ϕξθ+¨ϕ(η2ϕ−2ξs−3˙ϕξϕ)−2˙θ¨ϕξθ.

Inserting (16) - (19) into (13) and (14) yields

 [η1ss+˙θ(2η1sθ−ξss)+˙θ2(η1θθ−2ξsθ)−˙θ3ξθθ+2˙ϕη1sϕ+˙ϕ2η1ϕϕ +2˙θ˙ϕ(η1θϕ−ξsϕ)−2˙θ2˙ϕξθϕ−˙θ˙ϕ2ξϕϕ+¨θ(η1θ−2ξs−3˙θξθ) −2¨θ˙ϕξϕ+¨ϕη1ϕ−˙θ¨ϕξϕ]−sin2θ˙ϕ[η2s+˙ϕ(η2ϕ−ξs)+˙θη2θ−˙ϕ2ξϕ −˙θ˙ϕξθ]−η1cos2θ˙ϕ2|E1=0=E2=0, (20)
 [η2ss+2˙θη2sθ+˙θ2η2θθ+˙ϕ(2η2sϕ−ξss)+˙ϕ2(η2ϕϕ−2ξsϕ)−˙ϕ3ξϕϕ +2˙θ˙ϕ(η2θϕ−ξsθ)−˙θ2˙ϕξθθ−2˙θ˙ϕ2ξθϕ+¨θη2θ−¨θ˙ϕξθ+¨ϕ(η2ϕ−2ξs−3˙ϕξϕ) −2˙θ¨ϕξθ]+2cotθ(˙ϕ{η1s+˙θ(η1θ−ξs)−˙θ2ξθ+˙ϕη1ϕ−˙θ˙ϕξϕ} +˙θ{η2s+˙ϕ(η2ϕ−ξs)+˙θη2θ−˙ϕ2ξϕ−˙θ˙ϕξθ})−2˙θ˙ϕcsc2θη1|E1=0=E2=0. (21)

Inserting the values of and from the geodesic equations (11), and then comparing the coefficients of the powers of and , we obtain the following system of partial differential equations.

 (˙θ˙ϕ)0 : η1ss=0, (22) η2ss = 0, (23)
 ˙θ:2η1sθ−ξss=0, (24) η2sθ+cotθη2s=0, (25)
 ˙ϕ:2η1sϕ−sin2θη2s=0, (26) 2η2sϕ−ξss+2cotθη1s=0, (27)
 ˙θ2:η1θθ−2ξsθ=0, (28) η2θθ+2cotθη2θ=0, (29)
 ˙ϕ2:η1ϕϕ+sinθcosθη1θ−η1cos2θ−sin2θη2ϕ=0, (30) η2ϕϕ−2ξsϕ+sinθcosθη2θ+2cotθη1ϕ=0, (31)
 ˙θ3:ξθθ=0, (32)
 ˙ϕ3:−ξϕϕ−sinθcosθξθ=0, (33)
 ˙θ˙ϕ:2(η1θϕ−ξsϕ)−sin2θη2θ−2cotθη1ϕ=0, (34) (η2θϕ−ξsθ)+cotθη1θ−csc2θη1=0, (35)
 ˙θ˙ϕ2:−ξϕϕ−sinθcosθξθ=0, (36) −2ξθϕ+2cotθξϕ=0. (37)

Equations (32), (33) and (36) yield

 ξ=b1(s). (38)

Substituting the value of into (28) and solving with (22), we get

 η1=[f1(ϕ)s+f2(ϕ)]θ+f3(ϕ)s+f4(ϕ). (39)

From (23), (25) and (29) we obtain

 η2=−cotθf6(ϕ)+f7(ϕ). (40)

Substituting the values of and into (26) we get

 η1=[As+f2(ϕ)]θ+Bs+f4(ϕ). (41)

Then (24) yields

 ξ=As2+c1s+c0. (42)

Now (30) and (34) give

 ξ=c1s+c0, (43)
 η1=c3cosϕ+c4sinϕ, (44)
 η2=cotθ(c4cosϕ−c3sinϕ)+c2. (45)

Therefore, the exact symmetries come out to be

 X0 = ∂∂s,X1=s∂∂s,X2=∂∂ϕ, X3 = cosϕ∂∂θ−cotθsinϕ∂∂ϕ, X4 = sinϕ∂∂θ+cotθcosϕ∂∂ϕ. (46)

In the next sections we discuss two different approaches for finding approximate symmetries of this 2-manifold.

### 2.1 Approximate symmetries: First approach

Here we will convert the geodesic equations (11) of the sphere into perturbed equations by adding a general function and then try to find the approximate symmetries of these equations.

The second prolongations of the five exact symmetries (46) of the geodesic equations of the sphere are given below.

(i)

Here therefore, equations (16) -(19) give , so that the second prolongation becomes

 X0=∂∂s.

(ii)

Here , therefore, equations (16) -(19) give , , so that the second prolongation becomes

 X1=s∂∂s−˙θ∂∂˙θ−˙ϕ∂∂˙ϕ−2¨θ∂∂¨θ−2¨ϕ∂∂¨ϕ. (47)

(iii)

Here , therefore, equations (16) -(19) give , so that second prolongation becomes

 X2=∂∂ϕ. (48)

(iv)

Here , therefore, equations (16) -(19) give

 η1,s = −˙ϕsinϕ, η2,s = ˙θ(sinϕcsc2θ)+˙ϕ(−cotθcosϕ), η1,ss = −˙ϕ2cosϕ−¨ϕsinϕ, η2,ss = ˙θ2sinϕ(−2csc2θcotθ)+˙ϕ2cotθsinϕ+2˙θ˙ϕ(cosϕcsc2θ) +¨θcsc2θsinϕ+¨ϕ(−cotθcosϕ),

so that the second prolongation becomes

 X3 = cosϕ∂∂θ−cotθsinϕ∂∂ϕ−˙ϕsinϕ∂∂˙θ+(˙θsinϕcsc2θ−˙ϕcotθcosϕ)∂∂˙ϕ (49) −(˙ϕ2cosϕ+¨ϕsinϕ)∂∂¨θ+(−2˙θ2sinϕcsc2θcotθ+˙ϕ2cotθsinϕ +2˙θ˙ϕcosϕcsc2θ+¨θcsc2θsinϕ−¨ϕcotθcosϕ)∂∂¨ϕ.

(v)

Here , therefore, equations (16) -(19) give

 η1,s = ˙ϕcosϕ, η2,s = −˙θcsc2θcosϕ−˙ϕcotθsinϕ, η1,ss = −˙ϕ2sinϕ+¨ϕcosϕ, η2,ss = ˙θ2(2csc2θcotθcosϕ)+˙ϕ2(−cotθcosϕ)+2˙θ˙ϕ(csc2θsinϕ) −¨θcsc2θcosϕ−¨ϕcotθsinϕ,

so that the second prolongation becomes

 X4 = sinϕ∂∂θ+cotθcosϕ∂∂ϕ+˙ϕcosϕ∂∂˙θ−(˙θcsc2θcosϕ+˙ϕcotθsinϕ)∂∂˙ϕ (50) +(¨ϕcosϕ−˙ϕ2sinϕ)∂∂¨θ+(2˙θ2csc2θcotθcosϕ+˙−ϕ2cotθcosϕ +2˙θ˙ϕcsc2θsinϕ−¨θcsc2θcosϕ−¨ϕcotθsinϕ)∂∂¨ϕ.

Now, we approximate the geodesic equations of a sphere by using two arbitrary functions , and a small parameter as

 E11 : ¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0, E21 : ¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0, (51)

where

 E10 : ¨θ−sinθcosθ˙ϕ2=0, E20 : ¨ϕ+2cotθ˙θ˙ϕ=0, (52)

are the exact geodesic equations. For Case (i) the auxiliary functions become

 H1 = 1ϵX00(¨θ−sinθcosθ˙ϕ2+ϵf(θ))∣¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0 = 1ϵ(∂∂s)(¨θ−sinθcosθ˙ϕ2+ϵf(θ)∣¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0 = 1ϵ(0)=0,
 H2 = 1ϵX00(¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ))∣¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0 (53) = 1ϵ(∂∂s)(¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ))∣¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0 = 1ϵ(0)=0.

Hence , so we cannot proceed further.

Now we take Case (ii). Here the auxiliary functions become

 H1 = 1ϵX10(¨θ−sinθcosθ˙ϕ2+ϵf(θ))∣¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0 = 1ϵ(s∂∂s−˙θ∂∂˙θ−˙ϕ∂∂˙ϕ−2¨θ∂∂¨θ−2¨ϕ∂∂¨ϕ) × (¨θ−sinθcosθ˙ϕ2+ϵf(θ))∣¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0 = 1ϵ(−2¨θ+2sinθcosθ˙ϕ2+ϵ(0))∣¨θ−sinθcosθ˙ϕ2+ϵf(θ)=0 = 1ϵ(−2(−ϵf(θ)))=2f(θ),
 H2 = 1ϵX10(¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ))∣¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0 (54) = 1ϵ(s∂∂s−˙θ∂∂˙θ−˙ϕ∂∂˙ϕ−2¨θ∂∂¨θ−2¨ϕ∂∂¨ϕ) × (¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ))∣¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0 = 1ϵ(−2¨ϕ−2cotθ˙θ˙ϕ−2cotθ˙θ˙ϕ+ϵ(0))∣¨ϕ+2cotθ˙θ˙ϕ+ϵg(ϕ)=0 = 1ϵ(−2(−ϵg(ϕ)))=2g(ϕ).

Solving for in

 X1E10 ∣ E10=0+H1=0, X1E20 ∣ E20=0+H2=0, (55)

gives the same equations as (24)-(37) with a change in (22) and (23) given by

 η1ss+2f(θ) = 0, (56) η2ss+2g(ϕ) = 0. (57)

Equations (32), (33) and (36) yield

 ξ=b1(s). (58)

Substituting the value of into (28) and solving with (56), we get

 η1=f1(θ,ϕ)s+f2(θ,ϕ)−f(θ)s2. (59)

Integrating (59) twice with respect to , and using (28), we get

 η1θθ=f1θθ(θ,ϕ)s+f2θθ(θ,ϕ)−fθθ(θ)s2=0.

Comparing the coefficients of the powers of , we obtain

 f1θθ=0,f2θθ=0,fθθ=0, (60)

which implies that and are linear in Hence we can write

 f1(θ,ϕ) = f1(ϕ)θ+f3(ϕ), f2(θ,ϕ) = f2(ϕ)θ+f4(ϕ), f(θ) = cθ+d, (61)

so that (59) becomes

 η1=[f1(ϕ)s+f2(ϕ)]θ+f3(ϕ)s+f4(ϕ)−(cθ+d)s2. (62)

From (57), (25) and (29), we obtain

 η2s=0, (63)

therefore (57) gives

 η2ss+2g(ϕ)=0,

so that

 g(ϕ)=0, (64)
 η2=−cotθf6(ϕ)+f7(ϕ).

Substituting the values of and into (26) we get

 η1sϕ=0,f1(ϕ)=A,f3(ϕ)=B,

which gives

 η1=[As+f2(ϕ)]θ+Bs+
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