Etymonline has this to say:
-ics
in the names of sciences or disciplines (acoustics, aerobics, economics, etc.) it represents a 16c. revival of the classical custom of using the neuter plural of adjectives with -ikos (see -ic) to mean "matters relevant to" and also as the titles of treatises about them. Subject matters that acquired their names in English before c.1500, however, tend to remain in singular (e.g. arithmetic, logic).
So yes, at some point in history, there were such things as physic (meaning "natural science"), mathematic (meaning "mathematical science"), etc. that were later turned into plural forms but kept being treated as singular.
Edit: having looked in a few more places, it appears that in contemporary English, it still makes some sense to have both the suffix -ic and its plural form -ics. According to the Collins English Dictionary, the former has kind of specialized in forming adjectives, while the latter is happily forming nouns:
-ic
suffix forming adjectives
- of, relating to, or resembling: allergic, Germanic, periodic. See also -ical.
[...]
[from Latin -icus or Greek -ikos; -ic also occurs in nouns that represent a substantive use of adjectives (magic) and in nouns borrowed directly from Latin or Greek (critic, music)]
[...]
-ics
suffix forming nouns (functioning as singular)
- indicating a science, art, or matters relating to a particular subject: aeronautics, politics
- indicating certain activities or practices: acrobatics
[plural of -ic, representing Latin -ica, from Greek -ika, as in mathēmatika mathematics]
The key here is that they are not just two unrelated suffixes. Much rather, one is etymologically a plural form of the other. As the American Heritage Dictionary succinctly puts it, -ics is "-ic + -s".
There are two related problems here. First, the idiom one of the X always has a plural noun as X, even when there is an intervening phrase or modifier, and even if that modifier seems to want a singular head, as is the case with your One of the most common, and generally worst problem[s]. So the simple answer is that problems is correct in both of the sentences given above.
The related problem that you alluded to is the fact that the singular verb is occurs in close proximity to the plural noun problems. This is misleading: the number of the verb is determined by the number of the verb's subject, and the subject of the verb in your examples is One. The fact that one is modified by a prepositional phrase with a plural object does not change the number of the subject.
(There are some exceptions to the rule given above, but none of those exceptions apply to the phrase one of.)
Best Answer
Sometimes the usage is arbitrary, but there are two main tendencies in use: is the writer referring to the scientific field concerned with forces, or are they referring to those forces directly? Are they a scientific speaker, where the choice really matters, or are they a casual speaker?
Dynamics (or Dynamics) takes a singular verb when it refers to the subject of study. Compare to physics or (in some dialects) maths. For example:
However, when one is referring to forces directly, without concern for a wider body of study, then dynamics takes a plural noun and refers to multiple forces acting simultaneously. The Collins Dictionary specifies this usage in both British and American English. The American entry is as follows:
So this would hold true for the discussion of forces in a non-physics context, like in biology:
This is the general usage I'd expect in non-scientific contexts too. Your average speaker will tend not to think of "dynamics" as a singular entity. Specific searches of the Corpus of Contemporary American English supports this: "the dynamics is" has 28 hits, and "the dynamics are" has 105 hits, and "dynamics are" is in general more common.
That said, this line can seem fuzzy; I have seen dynamics used in cases where it could refer to multiple forces or to the study of forces:
Technical language is fun. At this point, it's up to authors or editors to determine what they mean and apply verbs accordingly.
A final nota bene: Collins claims that UK and American speakers differ in whether the musical term dynamics is treated as singular or plural. In the UK entry, it is treated as singular; in the American entry, it is treated as plural. Frankly I've seen both forms used online interchangeably, so I don't know how reliable that distinction is.