_{1}

^{*}

In polar coordinate system, we consider fifteen classes of forces resulting in unlimited undiscovered orbitals. The classic conic orbits are one of the special subclasses of the fifteen classes. Among the rest of the forces, we show a few instances displaying typical fresh orbitals. Aside from the common theoretical foundation, the specifics of the orbitals are given by the solution of corresponding equations of motion. These are coupled nonlinear differential equations. Solving these equations numerically, utilizing a Computer Algebra System such as Mathematica is conducive to the orbits. Simulation of the orbitals provides a visual understanding about the motion under the influence of the generalized noncentral forces.

In our previous work, we investigated the motion of a massive point-like particle under the influence of semi generalized central forces [_{ij} and g_{ij}. The main objective of our investigation is to apply the classical mechanics analyzing the orbitals of a massive point-like particle undergoing the influence of tabulated forces in

Following the objectives outlined in the previous section we consider the kinematics of a mobile massive point- like object of mass m in a two-dimensional space. Utilizing the polar coordinate system the acceleration is [

where according to the standard convention a single-dot and a double-dot are the first and the second derivatives with respect to time, respectively. Applying (1) the equation of motion is,

The RHS of (2) is one of the fifteen cells of _{12}; namely

The set of equations given in (3) are coupled ODEs. For arbitrary functions such as

In this section according to

Example 1. Consider the well-known gravity and electrostatic force i.e. the Keplerian forces. The force falls

F | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

4 |

in the category of cell_{12} of

stance, in the case of gravity the value of

constant and M is the central mass. In the case of charge-charge interaction

tric coupling constant and Q, q are the charges of the point-like charges. By trial and error the initial conditions and the force constant are adjusted so that the orbital is a perfect stable circle. One such set of parameters is given in the figure caption of

By adjusting the initial radial velocity the stable circular orbit of

The left plot of

In this case too our current output coincides with our previous work [

Example 2. Here we consider an example associated with the cell_{13}. We equate

For a set of parameters specified in the figure caption of

Description of the individual panel is the same as in the previous examples. It is worthwhile noting unlike the previous examples the orbital is not stable. The character of the pure radial force makes the particle orbiting about the center and then wandering away, tracing a non-returnable trajectory. For a better descriptive word maybe in this case “orbital” should be called “trajectory.”

Example 3. Here we consider an example associated with the cell_{14}. We equate

is the force constant. The spirit of this theoretical suggested force is similar to the previous example. Meaning, no such force has been observed in nature, yet! However, as mentioned before, the analysis paves the road for the “what-if scenarios.” Similar to the previous examples the angular momentum of the particle is conserved. For a set of parameters specified in the figure caption of

According to the plot of the middle panel the radial distance of the particle is a diminishing oscillatory function with respect to time, t. Its polar plot shown in the right panel is an interesting unstable orbital. Here the particle interestingly orbits about the center and contrary to example 2 stays in sight.

Example 4. As a last example we consider a case associated with cell_{32}. We equate

individual force strength x along the radial and the azimuthal directions. This example, unlike the previous ex-

amples, doesn’t conserve the angular momentum. For a set of parameters specified in the figure caption of

The orbital shown in the right panel is somewhat interesting. The rest of the plots are self-explanatory.

Motion of a particle under the influence of conventional forces such as gravity and electrostatic is known [

Our approach is purely numeric and is based on numeric solution of differential equations provided by Mathematica. We include a Mathematica code so that the interested reader is able to apply the code investigating the orbitals/trajectories of a particular force.

The author appreciates the referee’s comments.

Mathematica code. Here is the code that runs Example 4 in the text. In this example the right hand sides of (3) are −xr(t) and −xq(t), respectively. However, the given code can run for any desired case. To do so in the first line of the code one needs to replace the right hand side of the corresponding equations with the appropriate functions.