Sometimes, I have ideas for research in mathematical subjects about which I don't know much. Let me describe an example to make it more concrete.
In his 2009 article "The Brachistochrone Problem for a Disk" by L. D. Akulenko, he describes along which curve a disk with radius $R$ rolls down in the shortest amount of time. The problem that was considered is a generalization of the classical Brachistochrone problem, which entails finding the curve of quickest descent for a point mass.
In Akulenko's article, the density of the disk is assumed to be uniform. I wonder: what if we start tweaking this assumption? We could consider, for instance, the following three variations:
(Please excuse my bad drawing skills in this case. All circles have radius $R$.)
Variation $(A)$ concerns a disk rolling down a curve, for which the density is uniform except at an extra point mass at the edge. The question becomes: how does this extra point mass affect what the curve of quickest descent looks like? How can this curve be described mathematically, also as a function of the magnitude of the extra mass?
In variation $(B)$, the extra mass is put in a pie slice in the circle, and in variation $(C)$, it is put in a vertical segment of it. For all of these variations, the question remains the same: what are the curves of steepest descent?
I do not possess the required background on the theory of dynamical systems and/or (partial) differential equations to solve this problems. At the same time, I believe these who does have sufficient knowledge in these areas might be interested in doing research on these topics. Especially since, I believe, these particular generalizations of the Brachistochrone problem have not been considered yet (for other generalizations, see for instance this paper by Gemmer).
I find the idea that others might build on the questions I've raised appealing, because it could mean you've indirectly added a little bit of extra knowledge to the existing body of work on mathematics. Moreover, you've aided people with finding a research topic that suits them.
- Are there journals that welcome instances of people sending their ideas for research projects (which they can't or won't delve into themselves) and publish them?
- Are there any repositories - either online or on paper - specifically dedicated to receiving and listing ideas for projects that researchers might consider investigating?
- Are there other things one might consider doing with such ideas? For instance, is it wise to send them to professors in relevant research areas?