J. Nig. Soc. Phys. Sci. 4 (2022) 957

Journal of the
Nigerian Society

of Physical
Sciences

The Use of Differential Forms to Linearize a Class of Geodesic
Equations

J. M. Orverema,b,∗, Y. Harunaa, B. M. Abdulhamida, M. Y. Adamua

aDepartment of Mathematical Sciences, Abubakar Tafawa Balewa University Bauchi, Bauchi State, Nigeria
bDepartment of Mathematical Sciences, Federal University Dutsin-Ma, Katsina State, Nigeria

Abstract

Lie was the first to consider linearization of differential equations many years ago. Since then, a great deal of research has been done on lineariza-
tion of differential equations using various methodologies. Surprisingly, there has not been much progress in linearizing geodesic differential
equations. In particular, the use of differential forms to linearize a class of geodesic equations is not documented in the literature. Differential
forms are used to linearize a class of geodesic differential equations in this research. Geodesics on a plane, geodesics on a cone, and geodesics
on a sphere are examples. The solutions to these equations were discovered during the linearization process, as the findings of this study are
distinctive, innovative, and original.

DOI:10.46481/jnsps.2022.957

Keywords: Differential forms, Linearization, Geodesics equations, Ordinary differential equations, Second order

Article History :
Received: 27 July 2022
Received in revised form: 31 August 2022
Accepted for publication: 01 September 2022
Published: 10 October 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: J. Ndam

1. Introduction

A geodesic is a curve that minimizes length locally. It is, in
other words, a path that a particle that is not accelerating would
take. Geodesics are straight lines in the plane. Geodesics are
great circles on the sphere (like the equator). The term geodesic
refers to the curve that would be formed if you continued on
a straight path. You could travel a great circle on the surface
of a sphere (think of the earth) if you kept traveling straight
without turning left or right. Geodesics are the shortest distance
curves on a surface. They are useful in the transportation of
products and persons at low cost of time and energy, in addition

∗Corresponding author tel. no:
Email address: orveremjoel@yahoo.com (J. M. Orverem)

to their intrinsic interest. They are also critical as emergency
exit routes during flights. The methods of differential geometry
can be used to locate geodesics. The equator and the other great
circles on a sphere are common examples. In a curved space,
a geodesic is the straightest path conceivable. A straight line is
what we call space that is flat.

When you try to move straight in a curved space, you get
geodesics in general. For example, while you are driving on the
highway (and the road appears to be quite straight to you), you
are not driving on a straight line; instead, you are driving on the
straightest line conceivable on the Earth’s curved surface. Ask
what curve within the surface connects two neighboring places
on the surface and has the smallest length. A geodesic curve is
a curve that meets the condition. In a curved space, though, it
is just the opposite of straight. It’s a mathematical extremum,

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Orverem et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 957 2

which means that any slight divergence from it will make it
longer. The geodesic between two places on the Earth’s sur-
face is thus a great-circle path; nevertheless, the Earth’s surface
is curved locally as well as globally, that is, it has mountains
and valleys. As a result, more than one geodesic can exist be-
tween two places. Although they are not always the same length
(though they can be), they are both extrema.

Analysis of the geodesics of the Einstein-Maxwell-dilaton
theory’s exact solution, the four-dimensional linear dilaton black
hole (LDBH) spacetime was examined in [1]. The conventional
Lagrangian approach is used to investigate the test particles’
geodesic movements. They demonstrated that exact analytical
solutions to the radial and angular geodesic equations may be
achieved after obtaining the Euler-Lagrange equations. In par-
ticular, it is demonstrated that one of the radial trajectory solu-
tions may be expressed in terms of the Weierstrass P-function,
an elliptic-type special function. In another development as can
be seen in [2], some partial differential equations which were
derived from the well-known two logistic distribution parame-
ters were solved using two alternative techniques. The first ap-
proach was the conventional one, which required the solution
of three partial differential equations. The well-known Dar-
boux Theory was the second strategy. It was discovered that,
the geodesic equations are two minimum or isotropic curves.
Both approaches produced the same outcome, as was to be ex-
pected.

In the article [3], the differential pursuit game problem was
examined, in which a finite number of pursuers chase a finite
number of evaders. The problem is expressed in a Hilbert space
l2 with the motions of the pursuer and evader being defined
by nth and mth order differential equations, respectively. Inte-
gral and geometric restrictions are placed on the control mech-
anisms used by the evader and pursuers, respectively. The Lie
subalgebras of Noether symmetries associated with systems of
geodesic equations are found to have one-dimensional optimum
systems [4]. They also discovered invariants for every com-
ponent of the deduced optimal system. It was demonstrated
that the resulting invariants can transform systems of geodesic
equations—nonlinear systems of quadratically semi-linear second-
order ordinary differential equations into nonlinear systems of
first-order ODEs.

The article [5] demonstrate a theorem that connects the met-
ric collineations and the Lie symmetries of the geodesic equa-
tions in a Riemannian space. To Einstein spaces and spaces
with constant curvature, the findings were applied. The utiliza-
tion of the results is then demonstrated using examples. In [6],
group hydrodynamical systems with right invariant L2 or H1

metrics that may be expressed as geodesic equations on diffeo-
morphism groups or on extensions of diffeomorphism groups
was considered. The numerical solution of Geodesic equations
was investigated in [7] where the initial value problem on an
oblate spheroid, the direct geodesic problem was solved nu-
merically using both geodesic and Cartesian coordinates. In
that study, the differential geometry theory is used to formulate
the geodesic equations. The initial value problem in question is
reduced to a system of first-order ordinary differential equations
that is solved numerically.

Not so much is done in the area of linearizing Geodesics
equations. The relationship between isometries and symmetries
of the system of geodesic equations was used in [8] to construct
criteria for second order quadratically and cubically semi-linear
equations and systems of equations. The geodesic deviation Ja-
cobi equation that addresses finite size effects caused by gravi-
tational tidal forces was considered in [9]. It was shown how the
Jacobi problem in any spacetime that admits entirely geodesics
that can be integrated can be solved. Invariant Wronskians for
the Jacobi system that are linear in the ’deviation momenta’
were derived by linearizing the geodesic equation and its con-
served charges, resulting in a set of integrated first-order dif-
ferential equations. The continuous hybrid numerical approach
is taken into consideration in the work of [10] to solve second
order initial value problems of ordinary differential equations
in general. Utilizing the power series as the basis function, the
method of collocation of the differential system resulting from
the rough solution to the problem was used.

A differential form, which includes differentials, is a quan-
tity that may be integrated. f (x)d x is the differential form of the
integral

∫ b
a

f (x) d x. This is because it is integrated over a one-
dimensional region or path, this differential form has degree
one. One-form refers to the differential form of degree one.
The differential forms was previously used to linearize some
important differential equations [11-13]. In this research, our
attention is focused on the linearization of a class of Geodesics
equations.

The class of Geodesics equations is obtained from a unified
equation

y′′ − 2
f ′ (y)
f (y)

y′2 − f ′ (y) f (y) = 0, (1)

as presented in [14]. Given that f (y) = y, equation (1) becomes

y′′ −
2
y

y′2 − y = 0, (2)

which describes the geodesics on a cone. For f (y) = d + y,
equation (1) is now

y′′ −
2

d + y
y′2 − d − y = 0, (3)

which describes the geodesics on a plane, where d is a constant.
Again, for f (y) = siny , equation (1) becomes

y′′ − 2coty y′2 − siny cosy = 0. (4)

Equation (4) describes the geodesics on sphere.
As previously mentioned, [14] addressed the class of Geodesic

equations (2) through (4). A characterization of Sundman lin-
earizable equations in terms of one auxiliary function and ODE
coefficients was the method adopted. A direct alternative method
for creating the first integrals and Sundman transformations was
provided by using this criterion to explicitly acquire the general
solutions for the first integral. Equation (4) was also resolved in
[15] using symmetry and integration techniques for differential
equations. Our results utilizing the differential forms method
are consistent with those obtained by other methods in every
circumstance. The approach we took in this case is simpler and
gets us to the findings more quickly.

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Orverem et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 957 3

2. Approach of Differential Forms

Linearization through differential forms entails that, the in-
vertible change of independent and dependent variables

X = F (x, y) and Y = G (x, y) ,
that will map the general second order nonlinear ordinary

differential equation

y′′ = f
(
x, y, y′

)
, (5)

into a linear equation, should necessarily be in the form

y′′ + f0 + f1y
′
+ f2y

′2
+ f3y

′3
= 0, (6)

and the coefficients f0, f1, f2, and f3 must satisfy the conditions

f0yy + f0
(

f2y − 2 f3x
)

+ f2 f0y − f3 f0x

+
1
3

(
f2xx − 2 f1xy + f1 f2x − 2 f1 f1y

)
= 0, (7)

and

f3xx + f3
(
2 f 0y − f1x

)
+ f0 f3y − f1 f3x

+
1
3

(
f1yy − 2 f2xy + 2 f2 f2x − f2 f1y

)
= 0. (8)

Once the conditions in equations (7) and (8) are satisfied, we
proceed to construct a 3 × 3 matrix

M = Pd x + Qdy, (9)

where P =
(

1
3

) 
−2 f1 −3 f0 3 f0y + 3 f0 f2

0 f1 2 f2x − f1y − 3 f0 f3
−3 0 f1

 ,
Q =

(
1
3

) 
− f2 0 2 f1y − f2x + 3 f0 f3
3 f3 2 f2 3 f3x − 3 f1 f3
0 3 − f2

 ,
and solve the equation

dr = Mr, (10)

where r =


U
V
W

, a special solution is usually sufficient for the
three components of r. We can also construct

K = U/W, L = V/W. (11)

Next, we construct the 2 × 2 matrix

Z =
[

(2K − f1) d x − Ldy f0d x + Kdy
−Ld x − f3dy Kd x + ( f2 − 2L) dy

]
,

and solve for R from the equation

dR = ZR, (12)

where R =
[

Fx
Fy

]
=

[
b
c

]
. Finally, we solve

dF =
[

d x dy
]

R; (13)

the two independent solutions will be taken as F and G. What
is given here is the summary of the method. For more detail
please see [12], [11] or [13].

3. Linearization of a Class of Geodesics Equations

The unified class of geodesic equations is represented by
equation (1). At this point, we want to linearize the three equa-
tions that make up the geodesic equations class.

3.1. Geodesics on a Cone

Equation (2) which describes the geodesics on a cone has
the coefficients f0 = −y, f1 = 0, f2 = −

2
y , f3 = 0, that satisfy

the linearizability conditions (7) and (8).

With Pd x =


0 yd x d x
0 0 0
−d x 0 0

 , and
Qdy =


2
3y dy 0 0

0 −43y dy 0
0 dy 23y dy

 ,
M = Pd x + Qdy becomes M =


2
3y dy yd x d x

0 −43y dy 0
−d x dy 23y dy

 .
With this situation, dr =


2U
3y dy + V yd x + Wd x

−
4V
3y dy

−Ud x + V dy + 2W3y dy

 , where
dr = Mr and r =


U
V
W

 .
If V = 0, then dU = 2U3y dy + Wd x, dV = 0 and

dW = −Ud x + 2W3y dy. Again, Ux = W, Uy =
2U
3y , Wx = −U

and Wy =
2W
3y . V = 0, U = y

2/3sinx and W = y2/3cosx are
given as a special solution. Therefore, equation (11) is now

K =
y2/3sinx
y2/3cosx

= tanx , L = 0,

and the matrix Z becomes

Z =
[

2tanx d x −yd x + tanx dy
0 tanx d x − 2y dy

]
,

and dR =
[

2btanx d x − cyd x + ctanx dy
ctanx d x − 2cy dy

]
.

From dR, we see that

db = (2btanx − cy) d x + ctanx dy,

and

dc = c
(
tanx d x −

2
y

dy
)
,

where bx = 2btanx − cy, by = ctanx , cx = ctanx and cy =
−

2c
y . Integrating

dc = c
(
tanx d x − 2y dy

)
, we have lnc + lny2 − lnsecx = k.

That is, cy
2

secx = e
k = k, and then

c = k
secx

y2
,

where k is a constant.

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Orverem et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 957 4

We notice that by = cx so that by = k
secx tanx

y2 . Integrating,
we have

b = −k
secx tanx

y
+ g (x) . (14)

When we differentiate equation (14) with respect to x, we get

bx = −k

(
secx + 2secx tan2 x

)
y

+ g′ (x) . (15)

But bx is also expressed as bx = 2btanx − cy, that is

bx = −2k
secx tan2 x

y
+ 2g (x) tanx − k

secx
y
.

Now comparing the above with equation (15) and simplifying,
one sees that

g′ (x) − 2g (x) tanx = 0.

Using the method of integrating factor with p (x) = −2g (x) tanx
and q (x) = 0, one obtains the integrating factor as 1sec2 x . There-
fore, g(x)sec2 x = m, that is g (x) = msec

2 x or g (x) = mcos2 x , where
m is another constant.

Substituting g (x) = msec2 x into equation (14), one obtains
that

b = −k
secx tanx

y
+ msec2 x .

For dF =
[

d x dy
] [ b

c

]
, you have that

dF =
(
−ksecx tanx

y
+ msec2 x

)
d x + k

secx
y2

dy.

On integration,

F = −ky−1secx + mtanx − ky−1secx ,

and finally, F = −2ksecxy + mtanx .
Taking the coefficients proportional to the constants k and

m to be the linearizing point transformation, we have

X = tanx , Y =
2 secx

y
.

With the transformation Y = c1 X + c2, one sees that
2 secx

y =

c1tanx + c2, that is
2 secx = c1ytanx + c2y is the solution of equation (2).

3.2. Geodesics on Plane
We now proceed to linearize the equation that describes the

geodesics on plane. The equation is designated as equation (3).
Equation (3) is in the form of (6) and its coefficients: f0 =
−d − y, f1 = 0, f2 =

−2
d+y, f3 = 0 satisfy the linearizability

conditions (7) and (8), and therefore, it is linearizable using
differential forms.

Now,

Pd x =


0 (d + y) d x d x
0 0 0
−d x 0 0

 ,

Qdy =


2dy

3(d+y) 0 0
0 −4dy3(d+y) 0
0 dy 2dy3(d+y)

 ,
so that equation (9) becomes

M =


2dy

3(d+y) (d + y) d x d x
0 −4dy3(d+y) 0
−d x dy 2dy3(d+y)

 ,
and dr =


2Udy

3(d+y) + V (d + y) d x + Wd x
−4V dy
3(d+y)

−Ud x + V dy + 2Wdy3(d+y)

 .
Letting V = 0 gives dU = 2Udy3(d+y) + Wd x, dV = 0, dW =

−Ud x + 2Wdy3(d+y), and Ux = W, Uy =
2U

3(d+y), Wx = −U, Wy =
2W

3(d+y).
The situation above is satisfied by a special solution

U = (d + y)
2
3 sinx and

W = (d + y)
2
3 cosx . Therefore K = tanx , L = 0. Con-

structing the matrix Z, one sees that

Z =
[

2tanx d x − (d + y) d x + tanx dy
0 tanx d x − 2dyd+y

]
,

and equation (12) is now

dR =
[

2btanx d x − (d + y) cd x + ctanx dy
ctanx d x − 2cdyd+y

]
.

This situation becomes

db = (2btanx − (d + y) c) d x + ctanx dy, (16)

and

dc = c
(
tanx d x −

2dy
d + y

)
. (17)

Integrating (17) above, we have

lnc = lnsecx − 2
∫

dy
d + y

+ k ,

so that

c(d + y)2

secx
= ek = k.

Making c the subject, we have

c =
ksecx

(d + y)2
, (18)

where k is a constant.
Again, from equation (16), bx = 2btanx − (d + y) c, by =

ctanx , cx = ctanx and cy = −
2c

d+y, where by = cx.
Since by = cx, we have that by = k

tanx secx
(d+y)2

, and on integra-
tion, we have

so that

b =
kytanx secx

d(d + y)
+ g (x) . (19)

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Orverem et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 957 5

Differentiating b with respect to x, we have

bx =
kysecx
d (d + y)

(
2tan2 x + 1

)
+ g′ (x) .

One notes that bx is also expressed as bx = 2btanx − (d + y) c,
that is

bx =
2kytan2 x secx

d (d + y)
+ 2tanx g (x) −

ksecx
d + y

.

Comparing the two expressions of bx and simplifying, one has

g′ (x) − 2tanx g (x) =
−kdsecx − kysecx

d (d + y)
.

This implies that

g′ (x) − 2tanx g (x) =
−ksecx

d
. (20)

Equation (20) has now been transformed into a first-order lin-
ear differential equation that can be solved using the integrating
factor 1sec2 x .

Multiplication of the integrating factor and equation (20)
gives

cos2 x g′ (x) − 2sinx cosx g (x) = −
kcosx

d
.

This becomes

cos2 x g (x) =
−k
d

sinx + m,

where m is another constant.
Simplifying, the equation above becomes

g (x) =
−ktanx secx

d
+ msec2 x ,

and the value of b from equation (19) is now

b =
kytanx secx

d(d + y)
−

ktanx secx
d

+ msec2 x .

Referring to equation (13), that is, dF =
[

d x dy
] [ b

c

]
, one

sees that

dF =
(

kytanx secx
d(d + y)

−
ktanx secx

d
+ msec2 x

)
d x

+
ksecx

(d + y)2
dy.

Integrating the equation above, you have

F = k
(

2ysecx
d(d + y)

−
secx

d

)
+ mtanx .

Therefore

X = tanx , Y =
(y − d)secx

d(d + y)

is the linearizing point transformation of equation (3). The gen-
eral solution can be readily expressed using the transformation
Y = c1 X + c2, as

y = d + c1d (d + y) sinx + c2d (d + y) cosx .

3.3. Geodesics on Sphere

Next to be considered is equation (4) that describes the geodesics
on a sphere. The equation has the coefficients f0 = −siny cosy , f1 =
0, f2 = −2coty , f3 = 0 that satisfied the linearizability condi-
tions (7) and (8).

The 3 × 3 matrix M = Pd x + Qdy is now

M =


2
3 coty dy siny cosy d x d x

0 −43 coty dy 0
−d x dy 23 coty dy

 ,
where Pd x =


0 siny cosy d x d x
0 0 0
−d x 0 0


and Qdy =


2
3 coty dy 0 0

0 −43 coty dy 0
0 dy 23 coty dy

 .
Now, dr =


2
3 Ucoty dy + V siny cosy d x + Wd x

−
4
3 V coty dy

−Ud x + V dy + 23 Wcoty dy

 , so that,
letting V = 0, we have that

dU = 23 Ucoty dy + Wd x, dV = 0 and dW = −Ud x +
2
3 Wcoty dy. Also, Ux = W, Uy =

2
3 Ucoty , Wx = −U and

Wy =
2
3 Wcoty .

A special solution U = sinx (siny )2/3, V = 0, W =
cosx (siny )2/3 satisfies the situation above. Construction of K
and L shows that K = sinx (siny )

2/3

cosx (siny )2/3
= tanx and L = 0. With K

and L, the 2 × 2 matrix Z becomes

Z =
[

2tanx d x −siny cosy d x + tanx dy
0 tanx d x − 2coty dy

]
,

and

dR =
[

2btanx d x − csiny cosy d x + ctanx dy
ctanx d x − 2ccoty dy

]
,

so that

db = 2btanx d x − csiny cosy d x + ctanx dy, (21)

and

dc = ctanx d x − 2ccoty dy. (22)

From equations (21) and (22), one sees that

bx = 2btanx − csiny cosy , by = ctanx , cx = ctanx ,

cy = −2ccoty .

Integrating equation (22), we have that

c =
ksecx
sin2y

, (23)

where k = ek is a constant.
One notes that by = cx, therefore,

by =
ktanx secx

sin2y
.

5



Orverem et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 957 6

Integrating the equation above, you see that

b = −ktanx secx coty + g (x) , (24)

for some function g (x) .
We can see that when we differentiate equation (24) with

respect to x, we get

bx = −kcoty tan
2 x secx − kcoty sec3 x + g′ (x) . (25)

But bx is also expressed as bx = 2btanx − csiny cosy that is

bx = −2ktan
2 x secx coty + 2tanx g (x) − ksecx coty .(26)

Now, comparing equations (25) and (26) and simplifying, we
have

g′ (x) − 2tanx g (x) = −ksecx coty
(
tan2 x − sec2 x + 1

)
.

Using the identity tan2 x + 1 = sec2 x , one finally has that

g′ (x) − 2tanx g (x) = 0. (27)

Solving equation (27) using the integrating factor 1sec2 x , one
sees that g (x) = msec2 x , where m is another constant.

Substituting g (x) into equation (24), we now have that

b = −ktanx secx coty + msec2 x .

Now,

dF =
(
−ktanx secx coty + msec2 x

)
d x +

ksecx
sin2y

,

and on integration, F = −2kcoty secx + mtanx . The linearizing
point transformation is now

X = tanx , Y = 2coty secx ,

and the solution of equation (4) is now 2coty = c1sinx +
c2cosx .

4. Conclusion

The linearization of geodesics on spheres, planes, and cones
is accomplished using the differential forms technique. These
equations have convincing solutions as obtained from the trans-
formation Y = c1 X + c2. These three geodesics equations are
considered under the umbrella geodesics equation. The solution
or result of equation (4) is similar to the one obtained in [15]
where symmetry and integration method was used. It is shown
that the equation to be linearized must have the form (6), and
the coefficients must satisfy the linearizability conditions (7)
and (8). The findings of this study are distinctive, innovative,
and original.

Acknowledgment

We thank the anonymous referees for the positive enlight-
ening comments, which have greatly helped us in making im-
provements to this paper.

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