## The Lost P function

01 July 2014ACT researchers rediscover the use of a powerful mathematical theory from the father of modern analysis, Karl Weierstrass, and use it to solve three previously unsolved problems in trajectory design and celestial mechanics.

The explicit solution to fundamental problems such as the motion of a spacecarft subject to a constant radial acceleration, the motion of a spacecraft subject to a fixed acceleration (Stark problem) or the two-body motion in the perturbed relativistic regime, so far eluded researchers. While these problems are known to be “simple” (or integrable in the jargon of dynamical system theory), formulas to express their solution explicitly were not known.

Trying to understand why this was the case, scientists from the ACT started re-exploring one of the greatest achievements of mathematics in the 19th century: the theory of elliptic functions. Surprisingly, a student excercise in a classic textbook of modern analysis [Whittaker (1927)] (Example 2, pag.454) provided a crucial piece of the puzzle. Interestingly, the excercise reported a theorem from Karl Weierstrass simply acknowledged in a note: “This result was first published in 1865 in an inaugural dissertation at Berlin by Biermann, who ascribed it to Weierstrass”. This crucial piece of information initiated a long search through several historical text books. It was worth the wait, embracing Weierstrass formalism allowed ACT reserachers to quickly obtain elegant solutions to a number of fundamental problems related to space engineering, planetary science and celestial mechanics. These results were published in a series of papers taking advantage of what ACT researchers now refer to as Weierstrass’ lost theorem.

While other tools such as numerical integration or series expansion can be (and are) used to study these problems, the possibility of having an elegant explicit solution is of significance. In fact, the new expressions allowed the ACT researchers to characterise new dynamical effects such as the long-term evolution of Mercury spin axis and improve algorithms for interplanetary trajectory optimisation.

Francesco Biscani, principal scientist of the project and a previous research fellow of the ACT now working with the Max Planck Institute, describes the status of this project as “only the beginning”. He explains, “we expect much more to come in the future. Currently we are working on an efficient computer implementation of Weierstrass ideas and on applying our methodology to other classical dynamically integrable system that are significant for celestial mechanics and space engineering in general”. Regardless, this exciting new work has already produced several notable results with hopefully more on the way.

### Contacts:

Advanced Concepts Team: Dario Izzo

### References

Izzo, D., and F. Biscani. 2016. “On the Astrodynamics Applications of Weierstrass Elliptic and Related Functions.” In AAS 16-279. 26th AAS/AIAA Space Flight Mechanics Meeting, Napa, California, USA. [link]

Biscani, F., and D. Izzo. 2016. “A Complete and Explicit Solution to the Three-Dimensional Problem of Two Fixed Centres.”

*Monthly Notices of the Royal Astronomical Society*455 (4): 3480–93. [link]———. 2014. “The Stark Problem in the Weierstrassian Formalism.”

*MNRAS, Monthly Notice of the Royal Astronomical Society*. [link]Biscani, F., and S. Carloni. 2013. “A First-Order Secular Theory for the Post-Newtonian Two-Body Problem with Spin - I. The Restricted Case.”

*Monthly Notices of the Royal Astronomical Society*428 (3): 2295. [link]