The present article brigs classical orbits simulated with Maxima software [1]. The idea is to show that this computational algebra system (CAS) can be used to create animations based on numerically generated data. We have chosen to generate classical orbits animations because they are very interesting to demonstrate dynamically the consequences of Kepler laws, a common theme in Physics classes. We hope our codes help teachers who wish to apply a dynamical and technological approach to their classes by developing animated simulations with a free CAS. Therefore, we let our code pieces available at each section below for the teachers who wish to modify the GIFs according to their students context or simply study these code pieces in order to learn how to create their own simulations. The mathematical details of the situations presented here can be found in the article [2].

Circular orbits

The computation of circular orbits does not demand a numeric approach. However they can be useful to test if the code is coherent with the formulas related to Kepler laws and also to be compared with orbits of nonzero eccentricity. The code of this GIF is available here.

Elliptical orbits

As an example of elliptical orbits, we bring the case in which the trajectory has an eccentricity ε = 0.5. Examples like this are very interesting to show intuitively the consequences of the second Kepler law. The code piece of such case can be found here.

Comparison of circular orbits

A comparison between circular orbits can help the students to visualize  intuitively the third Kepler law. We can see that the orbiting object which has the trajectory of smaller radius completes a revolution around the attractor in less time. The code piece of this situations can be downloaded here.

Comparison of circular and elliptical orbits

The fourth case brings the comparison of a circular and an elliptical orbit which have the same large semi-axis. Our animation shows that the period of the orbit does not depends on the eccentricity, but only on the value of the large semi-axis, dynamically showing the consequences of the third Kepler law. You can find the code piece of this case here.

Conclusions

We hope we have convinced the reader that Maxima software can be an excellent alternative to create didactic material for Physics classes. We think this article synthesizes well the fact the CAS has the tools necessary to deal with complex situations, including the ones which need a numeric approach. For more details about the Physics of these situations and explanations of the main points of the code pieces found here, please read our article described in the Ref. [2].

References

[1] MAXIMA, Maxima, A Computer Algebra System, available at https://maxima.sourceforge.io/. Visited in January 17 of 2025.

[2] E. S. de Oliveira, Orbit simulations with Maxima software, manuscript under review.