User:Bradley Klee/Physics Initiative
Physics is a subject like mathematics, where quantitative knowledge is easy to encode in a simple language of functions and numbers. The ultimate goal of the physics initiative is to give the OEIS new functionality as a science reference, and to reach a new audience of science students who may have zero initial interest in number theory or integer sequences. We can accomplish this objective by improving classification, content, and outreach.
Unfortunately, the Keywords do not include [phys] for sequences that can somehow be measured in a physics laboratory. Yet, the effort to classify [phys] sequences has already started on the wiki index page: The multi-faceted reach of the OEIS#Physics. The Keywords also do not include [dfin] for sequences with Differentiably Finite Power Series. It would be nice to have this keyword as many interesting [phys] sequences also belong to [dfin].
We can effectively classify a [phys] or [dfin] sequence by adding content to its encyclopedia page. References or links may point to articles or web pages discussing the sequence in context of physical theory or laboratory measurement. The encyclopedia page can link to a data repository (Cf. A038534). The examples section can also be utilized to explain a sequence in its physical context.
The format of the encyclopedia pages requires us to condense results and insights to short comments or simple formula. Proof that a particular sequence belongs to class [phys] or [dfin] may not be at all apparent. This is an opportunity to make external documents that link into the OEIS: blog posts, computer applications, research articles, maybe even a book!
Case Study: Hamiltonian Periods
The complete elliptic integral of the first kind is a good place to start. The [nice] sequence A002894 is the Hadamard product of [core] sequence A000984 with itself. Both A000984 and A002894 determine differentiably finite power series, but the later is more certainly a [phys] sequence. Since the days of Euler, we have known that the plane-pendulum period-energy function generates the integers of A002894.
Following Ramanujan, we also cross reference A002894 with similar functions A006480, A113424, and A000897. These pages list Hamiltonian generating functions, which help to explain membership to both [phys] and [dfin] classes. Essentially all four sequences arise from energy-dependent Hamiltonian oscillations. To prove class memberships, we prototype an algorithm that derives the differential defining relation from Hamilton's equations of motion, and publish at Wolfram Demonstrations. The same code also allows us to print off proofs for sequences A300058, A295870, and A303790.