Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be the Brachistochrone problem, the Isoperimetric problem, and the Chaplygin problem.
"Calculus of variations" seems an accepted umbrella term; at least, looking at the corresponding Wikipedia entry, you'll recognize that most problems in this class are of the type you are looking for:
- The Catenary shape
- The Brachistochrone problem
- Isoperimetric problems
- Geodesics on surfaces
- Minimal surfaces and Plateau's problem
- Chaplygin problem
- Mountain pass theorem
- Principle of least action
- Fermat's principle
As others have pointed out, "calculus of variations" is the umbrella term for the class of problems you mention. Let me add that arguably a bigger umbrella term is ``optimal control" that can be thought of as a generalization of calculus of variations problems in the sense that it can handle dynamic constraints. Sometimes, it is possible to intentionally cast the calculus of variations problems in optimal control form to gain insight about the solutions. I recommend the article 300 Years of Optimal Control: From the Brachystochrone to the Maximum Principle to get a feel about the big picture. The last section of that article argues why solving problems like Brachystochrone using optimal control may be preferred over solving the same using calculus of variations.