Applying Differential Forms and the Generalized Sundman Transformations in Linearizing the Equation of Motion of a Free Particle in a Space of Constant Curvature

Equation of motion of a free particle in a space of constant curvature applies to many fields, such as the fixed reduction of the second member of the Burgers classes, the study of fusion of pellets, equations of Yang-Baxter, the concept of univalent functions as well as spheres of gaseous stability to mention but a few. In this study, the authors want to examine the linearization of the said equation using both point and non-point transformation methods. As captured in the title, the methods under examination here are the differential forms (DF) and the generalized Sundman transformations (GST), which are point and non-point transformation methods respectively. The comparative analysis of the solutions obtained via the two linearizability methods is also taken into account.


Introduction
The equation of motion of a free particle in a space of constant curvature is a differential equation of second order. It finds applications in many areas as stated earlier. Such areas include spheres of gaseous stability, equations of Yang-Baxter, the study of fusion of pellets, the concept of univalent functions, and the static reduction of the second member of the Burgers classes, see (Nakpim & Meleshko, 2010) and (Karasu & Leach, 2009).
The method of differential forms which was first investigated by (Harrison, 2002), was used earlier in (Orverem, Tyokyaa & Balami, 2017) and applied in (Orverem, Azuaba & Balami, 2017) to investigate the possibility of linearizing equation (1). The GST method was first established in (Duarte, Moreira & Santos, 1994) and later, in (Mustafa, Al-Dweik & Mara'beh, 2013) where only the Laguerre form was investigated. Equation (1) was also stated in (Nakpim & Meleshko, 2010) with a view of linearization through the generalized Sundman transformations (GST), but its solution was not obtained. This research is novel because the complete solution of equation (1) is obtained with the aid of the two methods under consideration DF and GST.
Note that (Nakpim & Meleshko, 2010) presented the complete form of linearization through the GST. This method was also used by (Orverem, Haruna, Abdulhamid & Adamu, 2021) to linearized the essential Emden differential equation as the complete solution was obtained. equation of motion of a free particle in a space of constant curvature tagged as equation (1) above. Additionally, the comparison of the results of equation (1) from the two methods is also considered.

Linearization of Equation (1) via the Differential Forms
The starting point is a second-order ordinary differential equation We assume a point transformation given by the variables with a requirement that, 2 2 = 0.
Putting , , and into equation (10) and dividing by to convert the differential forms to functions, we have where the are given by Thus, it is necessary through the differential forms' procedure that equation (3) should be cubic in the first derivative obtained as equation (16).
Now, we define and as and replace , , and in the 1-forms in equation (15) in favour of the , and , obtaining We also note that We see that the 1-forms , , in equation (19) and in equation (20) are now expressed in terms of these four known functions and . The first three of these 1-forms can now be substituted into equation (13) on the various functions. If we do that, the first equation for , gives the equation which is nonlinear in K. The other equations give the results: and − 3 = − ( − 2 ) + 3 ( − 1 ) (23) which are also nonlinear. However, we can simplify the situation by defining new variables: so that from equation (18) = , = , and from equation (20) 3 The equation (26) gives expressions for and . The equation (21) after substitution for , gives an expression which is linear in , and . The equation (23) gives an expression which is also linear. The equation (22) gives a linear expression The integrability condition on equation (26) gives a linear expression Equations (29) and (30) can be solved for and . Thus we have expressions for all derivatives of , , and , all of which are linear and homogeneous in the same variables. That is We summarize all these relations in a fine matrix equation which is not zero since is a matrix. Substitution for in terms of and gives the condition This matrix condition in equation (36) reduces to the following conditions: ) ( 2 − 2 1 + 1 2 − 2 1 1 ) = 0, and 3 + 3 (2 0 − 1 ) + 0 3 − 1 3 + ( 1   3 ) ( 1 − 2 2 + 2 2 2 − 2 1 ) = 0.
To summarize, we note that the original differential equation is cubic in ′ presented in equation (16) with the coefficients satisfying equations (37) and (38). Now, we shall construct the point transformations properly. We will need , and therefore we need to solve equations (34). Once the equations are solved, we produce and from equation (25). To find the ( , ) and ( , ) which we are seeking, we revisit equations (9) and solve for , , and . Solution for and gives .
Solution for and , shows that they satisfy the same equation, so we will write only equations for the derivatives of . We note that + = − and − = , so, we can solve these equations for and . We can also substitute for and in terms of and . We finally get 2 .
We substitute for , , and ⁄ from equations (19) and (20) respectively in terms of the expressions obtained above, to with , and .
We now have two equations that can be expressed in matrix form as This linear equation set can be solved for . There will be two independent solutions, which can be taken as and as seen in equation (39). Integrability is guaranteed by setting = 0.
Finally, we can solve and the two independent solutions can be taken as and .
(42) One sees from equation (41) where is a constant. Differentiation of equation (43) with respect to gives We notice from equation (42) that, = − , and on substitution of from equation (43), we see that which indicates that = .
One can integrate equation (45) using the integration by parts to obtain On differentiation of equation (46) with respect to , one sees that We also note from equation (42) that Equating equations (47) and (48) and simplifying, we have which can be solved with the use of integrating factor. Solving equation (49) and further simplification, one arrives at Now, differentiating equation (52) with respect to we see that Now, equations (51) and (53) are similar. Therefore, one can simplify this situation to see that We truncate the last term of equation (54) since it is also the coefficient of the constant , and integrate the result by parts to have Therefore, equation (52) becomes Without loss of generality, we let − = −2 = = 1, and interchanging the variables of constant , we have as the linearizing point transformation. The result in equation (56) is in line with the one presented in (Mahomed, 2007) above.

Linearization of Equation (1) via the GST Method
The transformation that is defined to be ( ) = ( , ), = ( , ) , ( is referred to as the GST. It is essential to note that, equation (57) is a non-point transformation.
Equation (3) must take the form with the aid of equation (57) to be transformed into a linear ODE for some functions ( ), ( ) and ( ).
We can find the functions and by solving the equations that follow: The , and from equation (59) are examined from the equations Again, from equation (1), one sees that it is in the form of (58) with the coefficients 0 = 3 , 1 = 3 , 2 = 0. Hence, one has the expressions for 3 , 4 and 5 as defined earlier to be 3 = 3, 4 = 3 and 5 = 9 respectively.
Using the transformation = + , where and are constants, we see from equation (56) where = 1 − .
Comparing the two results in (56) and (75), one sees that the method of differential forms gives the linearizing point transformation (56) which on another transformation yields the solution (79), while the GST method readily yields the solution as seen in (75). Both of the solutions are correct in their respect, but the solution (75) is more convincingly, the general solution of the equation given in (1).
Researchers can also look out for other approaches possible, suitable for the solution of this very important equation. This will help broaden the study of equation (1) considered in this research.