Coulomb, Yukawa, and Hooke’s Oscillations

Oscillations due to three different forces in three areas of physics: electrostatic, nuclear, and mechanics, are analyzed. The electrostatic long-range Coulomb force has a different character than the nucleonic short-range Yukawa force. Both are different from the linear Hooke’s force. The equation of motion of each case is solved applying a Computer Algebra System (CAS). It is shown that these oscillations have similarities and differences. Phase diagrams of all three cases are compared.


Introduction
We consider three distinct areas of physics: mechanical, electrical, and nuclear.
We search a common scenario overlapping these areas, i.e., oscillations. In the area of mechanics, we consider a linear retractable force, namely a spring that is subject to Hooke's law [1]. The impact of this force on a massive object is linear oscillations. Compatible with forthcoming cases, a setup composed of a spring and an inclined is considered. For the electrostatic theme, we consider a force that is in proportion to the inverse squared distance between two point-like charges i.e., the Coulomb force [2]. A setup like the previous case, a paired of charges and an inclined are considered. This is conducive to non-linear oscillations. And for the nuclear theme, we consider a pair of interacting nucleons. The nucleonic interaction, the Yukawa force [3] is the strengthened version of the Coulomb force.
With these envisioned common settings, a forum is set to compare the characteristics of the induced oscillations.
This report is composed of three sections. In addition to Section 1, in Section 2, by applying the fundamental laws of mechanics, we craft the equations of motions for each of the three cases. Except for the first case where the equation of motion is a trivial analytic solvable ODE, the other two cases are nonlinear ODE. Applying a Computer Algebra System (CAS) specifically Mathematica [4], these are solved numerically. Plots of these solutions are compared characterizing the oscillations [5]. Additional auxiliary information is obtained by plotting their associated phase diagrams. Section 3 is the conclusions highlighting the lessons learned.

Calculation
We begin with the Coulomb force. Then we move to Yukawa force and then for a reference point we include the Hooke's force. A) Coulomb force [2]. To weaken the gravity force we place a charged marble q of mass m at the top of an incline. A second identical charged marble is fastened at the bottom of the ramp. By adjusting the inclination angle, we adjust the gravity force. The effective weight of the marble along the incline is mgsin(θ), g is the gravity acceleration. Figure 1 depicts the setup. The length of the incline, i.e., the initial distance between the masses is d.
Equations of motion for all three cases are formed applying Newton's law, The net F includes two terms: 1) the above-mentioned effective weight and 2) one of the case-related forces. For the Coulomb interaction, this is, where k is the electrostatic coupling constants, . For identical charges F c is repulsive.
B) Yukawa force [3]. The force between two nucleons is attractive and short-range. It is formulated as, In (2), g 2 is the nucleonic coupling constant, ξ is the scale parameter that controls the interaction range, and , respectively. We make two notes concerning F Y within the content of our aimed objectives. Acknowledging that 1) Yukawa force, when applied to nucleons within a nucleus, is attractive. And 2) its effective range is being controlled by the inverse mass of the virtual pion, ξ as such the effective range of (2) is within femtometers. For the objective of our investigation a) by inserting a minus sign the force becomes repulsive and, b) we assign values to ξ such that (2) becomes a long-range compatible with our objectives.
It is insightful comparing these two forces, i.e. F C and F Y vs. their effective range. For the sake of comparison in Figure 2 we set, kq 2 = g 2 . American Journal of Computational Mathematics   them is gaining strength. And that the larger the ξ the longer the effectiveness of the Yukawa force. As shown for a certain value of ξ, i.e., 0.6 the Coulomb and the Yukawa forces are closely comparable; at a certain separation distance, they cross over signifying the same strength. Concluding, Figure 2 suggests the resulting oscillations due to various values of ξ ought to be compared. This is reported in the upcoming paragraphs.
For the Coulomb force the equation of motion for the sliding marble is, In ( As previously noted for comprehensive comparison the strength of the forces, α, c in (3)(4) are set the same. Applying Mathematica [4] with initial conditions: plots of the solutions of (3-4) are shown in we display the corresponding phase diagrams. These are depicted in Figure 5.
As shown irrespective of the nonlinearity of the forces all three paired phase diagrams are closed. As expected, the first two pairs are somewhat alike. This is expected because the Coulomb and Yukawa forces have much in common, c.f. Figure 2. In these diagrams, their adjusted force parameters are counted for the aspect ratio. The last pair is self-descriptive.

Conclusion
We set an objective to compare the impact of the Coulomb and the Yukawa force for oscillations of paired massive point-like objects. The first force is somewhat familiar, the interaction is in proportion to 1/dist 2 , yet the character of the second one being confined to the short-range nucleonic force is confined to the mass of the exchanged mesons. No interest has been shown in the literature concerning their potential induced oscillations. In our investigation systematically we showed the similarities and differences of the impact of these forces. We have not applied quantum mechanical nonrelativistic Schrodinger physics, we strict our analysis to classical physics. The assumption is made concerning the impact of the Yukawa force beyond the short range. By solving the associated nonlinear equation of motion, we have shown the impact of these nonlinear forces on the classic two-body oscillations.