Let's get started!
In this sentence "get started" means that "to begin or start".But as far as I know "get + v-ing" has that meaning too .Such as in the following examples :
We have to get moving now.
It's getting dark.
So can we use "get+past participle" to mean "to begin or start" ? or "get started" is just an exception ? I mean can I say
I have to get written.
instead of
I have to get writing.
get started, get finished: Those are idiomatic.
Those mean: begin doing something or get through something, a task.
But get plus a gerund meaning to start has invaded other areas, too.
get writing, get studying, get moving, get going, get rolling, get dancing, [movement] Get here means: start, start writing, start studying, start going, get writing. It's often heard in AmE speech.
Get also means become: get rich, get tired, get bored, etc. Get is very complicated so I am limiting myself to those aspects.
As you say, "started" is a past perfect participle which effectively is an adjective,
The program is started.
It seems to me you can use the "get + past perfect participle" with any verb where the past perfect participle is a similarly idiomatic adjective.
We need to find a way to get them motivated.
You'd better leave before you get tired.
In the dark of the moonlight night the young explorers got turned around and didn't know where they were.
Or, more colloquially:
Come on Willy. Let's get sloshed!
The past perfect participle of "write" is "written", and it's perfectly fine to say something like:
Let's get this (report) written.
in the same way you might say:
Let's get this (project) done.
"Written" here is an adjective that describes the intended status of the report.
However you do have to be careful since the meaning of "written" can vary depending on context. A "written report" is usually considered to be one that has been written by hand not one that is completely finished. As usual, you have to know what is idiomatic.
"Get started" is a fixed phrase. It seems you are troubled by the thought that in the sentence "Let's get started on the essay" we is the subject and the agent of the action, the essay the object and patient of the action. You are right to think that "start" here is a past participle.
It might be easier for you to understand its grammar structure this way: it is true people idiomatically say "get started", but "get" can also be used in the structure get + object + past participle:
Let's get the job done tonight.
I will get you started on the project.
"Start" as a transitive verb can be used with a person (sentient being) as the object. For example, you can say:
The lecture started me thinking.
So such phrases as "get someone started" "get the work done" indeed imply passive voice. You can understand "let's get started" in a similar light. Subject + get + past participle means upon the subject an action is carried out. "He got dumped" means "he was dumped".
Think of it this way: in "get + past/present participle" the participle gives you a state. The subject of "get" will be put in/achieve that state through "get". For example, "He got tired" means he was put in the state of being tired. "It is getting dark" means it (the environment/sky) is reaching the state of being dark. By the same token, "I should get going" means I should reach the condition of moving.
