Dynamic modality: expressing ability and willingness

The word dynamic derives from the Greek dynamikos meaning strong or powerful. In English it has been defined as:

having a lot of ideas and enthusiasm; energetic and forceful
(Cambridge International Dictionary of English, 1995:434)

Linguistically, we need to be a bit more exact and define this form of modality as that which expresses ability or requirement of the subject to do something. It is non-subjective insofar as it refers to fact rather than likelihood or obligation (which are expressed via epistemic and deontic modality respectively).

You may find dynamic modality referred to as personal modality because it refers to individuals and their ability or willingness to do something. We don't use that term in this guide because dynamic modality may not, in fact, always refer to an individual, although it usually does, but it is an alternative.

Dynamic modality is one of the usual four categories of modality (the others being epistemic, deontic and alethic to which there are separate guides).
If these terms are mysterious to you, you should follow the guide to types of modality before tackling what follows.
There is, in fact, an argument that dynamic modality is not actually a form of modality at all because the speaker / writer is not:

using a modal expression to express an opinion regarding the truthfulness or necessity of an event, state or action
OR
affecting the current situation by what is said

For example, with the modal auxiliary verb can, four clear types of modality can be expressed:

Example	Type of modality expressed
You can go now	Deontic: giving permission / removing the obligation to stay
It can't be a rectangle if it has five sides	Alethic: expressing a factual state (also, arguably, not really a modal use at all)
That can't be the right answer	Epistemic: expressing the speaker's view of the truth of a proposition
She can play the piano very well	Dynamic: stating a simple fact about ability

Traditionally, following Palmer (2001) only the modal auxiliary verbs can/could and will/would express dynamic modality, i.e., express ability or willingness.
There are other ways to express the same notions, of course, and the grammar that is involved is not as straightforward as it might seem.
The following is an analysis which is focused on teaching the concepts and notions.

Expressing willingness

Both will and can are used to express the notion of willingness as in, for example:

Jane will drive you to the station if you like
Jane can drive you to the station if you like

Functionally, these two utterances are identical: they both express the subject's (Jane's) willingness to do something and do not express the speaker's view of any existing obligation or its lack or any comment concerning the truthfulness of a proposition.

There is, however, a difference in sense:
Sentence 1. concerns willingness alone but sentence 2. may be construed to suggest that not only is Jane willing to drive you to the station but that she is also able to do so (because, e.g., she has no other outstanding commitments).
The differences become clearer when set in a dialogue:

Fred:	Jane will drive you to the station if you like	Fred is stating that he understands that Jane is willing to do this
Jane:	Sure, no problem	Jane is confirming her willingness
OR
Jane:	Sorry, I won't drive in the dark	Jane is stating her unwillingness

Fred:	Jane can drive you to the station if you like	Fred is stating that he understands that Jane is willing to do this
Jane:	Sure, no problem	Jane is confirming her willingness
OR
Jane:	Sorry, I've got to take the kids to school	Now she is stating her inability rather than her unwillingness

The border between the notions of willingness and ability is blurred in English because we can express both functions with the same modal auxiliary, can/could, with no change to the intonation.
An example of this is a statement such as:
    I can help you with your homework
in which it is unclear (without a very explicit context) whether the speaker is stating an ability or a willingness to do something. It can mean, therefore, either:
    I am able to help you with your homework
or
    I am willing to help you with your homework
The same ambiguity occurs with the interrogative forms:
    Can you help me with my homework?
    Could you help me with my homework?
in which the questioner may be referring to the hearer's ability or willingness.

With the negative form, this ambiguity mostly disappears because:
    I can't help you with your homework
implies inability rather than unwillingness as does:
    I couldn't help her
In order to state unwillingness, we have to use the modal auxiliary will:
    I won't help you with your homework
    She wouldn't help her with her homework

will / would: the multi-faced verb

Much confusion can be avoided if the verb will is considered polysemous (i.e., having two or more meanings). In English the verb can, among other things, express

Futurity (a non-modal use) as in, e.g.:
1. I'll be 50 tomorrow
2. The train will arrive at platform 6
3. When will the train arrive?
4. I won't see him tomorrow
Certainty or likelihood (epistemic modality) as in, e.g.:
1. That'll be the postman
2. This'll be a long job
3. This won't take long
4. Will it rain?
Willingness (a dynamic modal use) as in, e.g.:
1. I'll help you
2. She'll read you a story
3. Will you marry me?
4. She won't tell me the truth

We are only concerned with the last of these here. These meanings are quite different and mixing up examples of the three functions in the same classroom presentation can bewilder learners unnecessarily.

The situation is similar with the verb in the past, would. It can, among other functions, signify:

Distancing and politeness
1. Choosing
      Would you help?
  over
      Will you help?
  or choosing
      If I give you the money will you buy it for me?
  over
      If I give you the money would you buy it for me?
  implies greater politeness and tentativeness although both refer to current willingness
Indirect speech
1. I'll do that for you
  is reported (at another place and time) as
  She said she would do it for me
  but both refer to her willingness
The habitual aspect
1. He would often complain
  expresses a habitual or repeated action and is not to do with (un)willingness at all.

Ambiguity

Because the verb will / would is polysemous (i.e., having more than one connected meaning), there is scope for ambiguity to arise.
For example, in:
    Will you marry me?
there is no ambiguity because the speaker is clearly referring not to ability but to willingness but in:
    Will he marry her?
the speaker may be referring to the future likelihood of him marrying her or to his willingness one way or the other. Without a context, we cannot know.
And in:
    He would be rude to her
the speaker may be referring to habitual actions, futurity in the past or to the likelihood that he may choose to be rude in the future. We cannot know without context.

Expressing ability

The verb can is often the first modal auxiliary verb, along with must, that learners encounter in English. It expresses a simple and fundamental concept and is, in the present tense at least, uncomplicated so we have, simply:

She can do that
She can't do that
Can you do that?
Can't you do that?

In the present tense, all these instances of can are replaceable with a copular verb (usually be) and the adjective able plus to:

She is able to do that
She isn't able to do that
Are you able to do that?
Aren't you able to do that?

although the use of the able-to formulation is less common and more formal. When no difference in meaning exists, English speakers will usually prefer the modal auxiliary.
Other copular verbs make subtle changes to the meaning but the sense remains:

She seems able to do that
She doesn't appear able to do that
She ended up able to do that
Did she prove able to do that?

That this formulation is analysable, and teachable, as copular verb phrase + adjective phrase + to-infinitive verb phrase (plus the object if any) is revealed by the parallel forms such as:

I am happy to do that (expressing willingness)
She is reluctant to do that (expressing unwillingness)
She is wholly unable to swim (expressing inability)
Isn't she required to be here (expressing a lack of obligation)

and so on.

So common is the concept that the verb can is often directly translatable into a range of languages and so fundamental is it that many languages simply attach a prefix or suffix to the verb to express the concept.
English is, however, unusual in that can is defective, and has no future form at all and only a restricted past form, could. The use of the adjective in be able to is the workaround in English so we get, e.g.:
Were you able to see the doctor?
I won't be able to come tomorrow
etc.

Expressing ability in the past

In the past tense, however, things become a little more difficult.

General and specific abilities

A general ability in the past can be expressed either with could or was / were able to:
I could speak French when I was 10 = I was able to speak French when I was 10
When I worked in London, I could catch the train to work = When I worked in London, I was able to catch the train to work
However, if we refer to a specific instance of success, only was / were able to is possible:
    He was able to say the word in French
NOT
    *He could say the word in French
Other specific instances of an ability can be expressed with other formulations, often verb processes such as:
    We succeeded in getting tickets
    They managed to catch the train
etc.
But not, usually:
    *We could get tickets
    *They could catch the train

Three exceptions

Verbs of perception can be expressed as abilities in the past with no sense of whether it is a general or specific ability to which we refer. E.g.:
    She was able to hear the telephone ringing = She could hear the phone ringing
    He was able to see the harbour = He could see the harbour
    I couldn't understand him = I wasn't able to understand him
With negative-sense adverbials. E.g.:
I was hardly able to hide my feeling = I could hardly hide my feelings
They were only able to find expensive tickets = They could only find expensive tickets
With subordinate clauses. E.g.:
They were delighted she could come = They were delighted she was able to come
He said he couldn't come before 6 = He said he wasn't able to come before 6

Fulfilled and unfulfilled ability

In the past, the perfect form of could implies an unfulfilled ability but the was / were able to formulation implies a fulfilled ability. For example:

I could have spoken to her on the phone
refers to the fact that I was able to do so but didn't for some reason (it may also mean that I consider it possible but have no recollection of doing so but that is epistemic modality and does not concern us here)
I was able to speak to her on the phone
refers to the ability and to the fact that I did speak to her
Could they have talked to her? / Couldn't they have talked to her?
both refer to whether they had the opportunity and the ability but assumes that they did not talk to her

In the negative, could have usually refers to possibility, rarely to ability, so it is not easily usable for dynamic modality. The statement:

They couldn't have talked to her
can only express epistemic modality (i.e., that it was impossible)
but:
I couldn't have driven any faster
does express ability and is often followed by something like (even) if I'd tried

Ambiguity

Because can and could are used for a variety of functions apart from dynamic modality concerning ability, there is often no easy way to disambiguate. For example:

Example	Possible modality
You can talk to him	Deontic: giving permission to talk to him Dynamic: expressing ability assumed in another
She could explain clearly	Dynamic: expressing her ability in the past Deontic: expressing the fact that she should explain Epistemic: expressing a future likelihood
You can't be serious	Dynamic: expressing frustration at another's inability Epistemic: expressing the speaker's view of the truth of a proposition
It could bend	Dynamic: stating a fact about the material Epistemic: stating a possible outcome
Can you be quiet?	Dynamic: Are you able to be quiet? Deontic: Please be quiet!
She could have driven faster	Dynamic: she had the ability to drive faster Deontic: she should have driven faster
You could have told me	Dynamic: you had the ability to tell me Deontic: you should have told me Epistemic: you may have told me but I don't remember

Only the context and understanding of speaker intentions along with some knowledge of the shared information can determine which meaning is appropriate.

Other ways to express dynamic modality

Because of the ambiguities which are rife with the two modal auxiliary verbs we have discussed, especially in positive clauses, whenever there is a need to disambiguate what is meant, native speakers will often resort to non-modal-verb ways to express the concepts of dynamic modality and (un)willingness or (in)ability.
There is a guide, linked below to non-modal-verb modality which contains more detail. Here some examples will suffice:

Expressing willingness:
1. Verbs
      I want to help
      She is offering to help
      They are proposing to give us a hand
      I refuse to help
      I don't want to come
  etc.
2. Adjectives
      I am willing to help
      She is keen to come
      I was eager to go
      They are reluctant to interfere
      I am loath to ask
      He is half-hearted about doing it
      I am delighted to help
      She is happy to lend us the money
      They are averse to eating out
  etc.
3. Adverbs
      They came willingly
      He accepted eagerly
      She arrived reluctantly
  etc.
4. Nouns
      He showed little commitment
      Her eagerness to meet him was obvious
      His reluctance to come was clear
  etc.
Expressing ability:
1. Verbs
      He was able to come
      They achieved the success they wanted
      She failed to win the prize
      I managed to do it
      It worked!
      He succeeded
      They persuaded me
      It demonstrated its value
      They attained the goal
  etc.
2. Adjectives
      It was ineffective
      That was a fruitful meeting
      It was a productive idea
  etc.
3. Adverbs
      He managed it successfully
      They worked effectively
      She operated productively
  etc.
4. Nouns
      His success was welcome
      The outcome was positive
      That was a fine accomplishment
  etc.

Many non-modal-auxiliary-verb ways of expressing dynamic modality are more formal versions and often used in writing in which context is limited and the scope for ambiguity is wider.

Related guides:
non-modal-verb modality	covers some of the above and extends it to include epistemic and other forms of modality
epistemic modality	which follows a similar format to this one
deontic modality	which follows a similar format to this one
multiple modalities	for a guide which considers how different types of modality may be combined in single utterances
ambiguity	for a guide which considers the polysemous nature of many modal auxiliary verbs and the ambiguity which can arise
the modality map	for more choices

References:
Palmer, F. R, 2001, Mood and Modality, 2nd edition, Cambridge: Cambridge University Press
Proctor, P (Ed.), 1995, Cambridge International Dictionary of English, Cambridge: Press Syndicate of the University of Cambridge