What should you do if you have a research idea outside your area of expertise?

Background

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.

Questions

1. 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?
2. Are there any repositories - either online or on paper - specifically dedicated to receiving and listing ideas for projects that researchers might consider investigating?
3. 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?
• It is very strange that the disk problem has not been considered before 2009. Sep 10, 2022 at 14:14
• Before the internet what I did was to look up all the references in the papers you have (at least those references that seem worth looking into), then the references of those references, and so on. And if any journals seemed to have several relevant papers, I'd go to the library and flip through the table of contents for volumes of that journal after the latest such paper. Then I'd go to Math. Reviews and look up some of the reviews of the most significant papers to see if the reviewer made any relevant comments. Also history books that treated the topic. Rinse, repeat, etc. (continued) Sep 10, 2022 at 16:15
• Now one can simply google. For instance, I found a large number of items to look at (nearly all of which are going to be freely available online) simply by googling "brachistochrone" in google-books, date restricted to 1800s to get things the authors of pre-internet papers may not have known of, especially those for an undergraduate U.S. "math club" devoted mainly to students at non-research colleges as is the case for one paper you cited). There is also the online Math. Reviews, if you have access. Sep 10, 2022 at 16:22
• The journal/periodical Pi Mu Epsilon Journal is for Pi Mu Epsilon, one of the two main U.S. -wide math club organizations I'm aware of (and whose journal I've cited many times online), the other being Kappa Mu Epsilon (whose journal is The Pentagon). (continued) Sep 12, 2022 at 7:21
• I used to look through library volumes of these a lot, beginning in the late 1970s, and have a large number of articles from them that I photocopied beginning around the mid 1980s when I started photocopying math papers of interest to me as I came across them in various college/university libraries. Thus I'm very familiar with Pi Mu Epsilon Journal, and while I find it useful for expository articles suitable for interested undergraduate math students, the absence of what you want in Gemmer's paper is not particularly significant (although it is more detailed than typical for this journal). Sep 12, 2022 at 7:31