There are three modal (or semi-modal) verbs of capacity, ability or capability in standard present-day English. These are can, be able to, and manage to.
All have past and negative forms. The irregular past tense of can in this sense is could.
The usages and subtly different meanings pertaining to these forms overlap and can be quite confusing. Manage to is a relative newcomer and helps to clear up the confusion.
Problem No. 1
Like most modal verbs, can is ‘defective’. It lacks verb forms other than the Present Simple and Past Simple. To make up for this, be able to is used to fill the missing forms.
 I can swim means I have the (permanent) ability to swim.
But, if we want to use the verb in the infinitive, we have to revert to be able to, thus:
 Every child should be able to swim.
Likewise, the –ing form.
 Being able to swim is a prerequisite for joining the marines.
Other forms of the verb need to be used less regularly in this context. But likewise require use of be able to.
 Present Perfect: I have been able to swim since I was a baby.
The language (by which I mean people using language over time) has found an elegant solution to this problem.
But then comes Problem 2:
The distinction between can and be able to is also used (in some contexts) to distinguish between a permanent and temporary capacity respectively.
This is subtle to the point of obscurity.
Back in the day, I used to pose conundrums like this:
 Algernon survived the shipwreck because he could swim.
Bertrand survived the shipwreck because he was able to swim.
Who can swim?
The answer is Algernon, but the question understandably gives rise to some confusion.
To make the answer uncontestable, we would have to ask:
If only one of them can swim, which one is it?
Can and be able to are too closely intertwined grammatically to make this distinction clearly and cannot do so at all in the case of forms of the verb in which can is defective.
This is where the relative neologism, manage to, used modally, comes in.
Manage to clearly takes over the temporary fumbling function of be able to and leaves the distinction uncontestable.
 He survived the shipwreck because he was able to swim.
 He survived the shipwreck because he managed to swim [‘even though, he couldn’t swim’ is implied]
This is a clear example from present-day English of what some contemporary linguists call ‘grammaticalization’.
This is an ugly and somewhat misleading term.
The idea is that all grammatical morphemes and lexemes originate in pure lexical terms, whose usage has become diluted over time. This may lead to distortions in the phonetic or written form of the word. For example, ‘gonna’ tends to mark the grammatical future use of the phrase ‘going to’ while its more literal lexical meaning, as in ‘I am going to London,’ still exists.
In this case, a completely different phonetic form has been spun off by the ‘grammaticalization’ process.
In other cases there is a change in syntactical structure.
Keep as a modal verb indicating repeated (usually undesired) action only in so far as it is followed by the –ing form of another verb.
 She keeps hiccoughing.
 He keeps tropical fish.
In these two sentences the verb ‘keep’ clearly has a very different meaning.
The same occurs with ‘manage’ in its modal capacity.
 He managed to stay afloat after the shipwreck, despite not being able to swim.
 He managed the shipping company for many years.
Why and when the word ‘manage’ (or ‘keep’ for that matter) came to take on this additional grammaticalized function is a much more interesting question.
The word ‘manage’ originally appears in neo-Latin languages in the Middle Ages to refer to horsemanship (the training and riding of a horse). By the late 16th century, its use in English had extended to any activity requiring manual dexterity and to the ‘management’ of a business in the modern sense. The extended modal use appears only in the mid 18th century and I suspect that its commonplace use in this sense is even more recent.
The ‘descent’ of this word from highly-specialized aristocratic equestrian skills to something as mundane as just ‘managing to get out of bed in the morning’ provides a lesson not only in language change but also in significant democratization that society has undergone in the past five hundred years or so.
It also contains a broader message about the ups and downs of linguistic change and an implicit warning.
As I suggested above, ‘grammaticalization’ is not a good word. It suggests that there is a natural downward flow from the particular to the general and that such ‘grammaticalized’ words always lose lexical force. Neither of these claims is true, as most ‘grammaticalization’ theorists freely admit.
I would suggest, on the contrary, that this seemingly universal language process, provides evidence of a constant process of ‘lexicalization’. As grammatical elements become increasingly eroded and the rules invented to govern them growingly abstruse, people naturally reach out for more clearly visualizable words to take their place.
I would also argue the lexical content of so-called ‘grammar words’ and even morphemes is almost never truly lost and always available for resuscitation. People may say ‘gonna’ when they mean it in the modal sense and ‘going to’ when they mean it in the literal sense. But they still know in their heart of hearts that the two are in fact the same. This is conscious or semi-conscious. But the effect may be unconscious too.
The modal verb ‘will’ for example has become heavily ‘grammaticalized’ in modern English. To the extent that in some old-fashioned grammar books it is presented simply as a marker of an English Future Tense similar to that of neo-Latin languages. More enlightened modern grammarians have entirely dismissed this notion. English has no future tense so to speak, only various forms used to express a multitude of shifting attitudes regarding the future. ‘Will’ is one of them, but its specific use depends directly in many cases on the original lexical meaning of the word (‘want’), arguably increasingly so.
To return to management. ‘Manage’ has been ‘dumbed down’ to some extent, but it has also been ‘dolled up’ in the form of the abstract noun ‘management’ to refer to a whole arcane industry of theorizing as to the skills required to best administer companies in a capitalist environment, most of which are far less specifiable than the eminently practical and verifiable skills required to rear and ride a horse.
While most of us struggle to manage to get out of bed in the morning and make ends meet, it is worrying that a term of obvious aristocratic pedigree is now being used (often mendaciously) to fabricate and consolidate a new soi disant upper stratum in an increasingly unequal society.