Difference between "keep something on", "hang something on", and "leave something on"

Question

I came across three verb + on phrases in today’s New York Times article written by Maureen Dowd and titled “Stripped off Dignity”; it dealt with unpleasant airport pat-down practices.

In the following paragraph, I find the phrases “Keep jacket on,” “Hang on to unfinished bottle,” and “Leave shoes on.” It appears to me all these “keep on,” “hang on” and “Leave on” simply mean “keep on.” It may seem a silly question to you, native English speakers, but I’m curious to know the exact difference in meaning between the above three phrases, and to also understand why does "leaving" shoes “on” means to keep wearing shoes?

John Pistole, the T.S.A. chief, said they are trying to move past a “one-size-fits-all” program and implement a “risk-based, intelligence-driven process” by the end of the year that would have more refined targeting. If passengers are willing to share the same information they give to airline frequent-flier programs, he said, maybe someday they will be able to keep their jacket on and their laptop in their briefcase and hang on to that unfinished bottle of water.” “I’d like to get to the point,” he said wistfully, “where most people could leave their shoes on. 

Answer 1

To "keep something on" and to "leave something on" mean virtually the same thing. Either can refer to an article of clothing that one intends to (or is admonished to) persist in wearing, or to some kind of powered machine (a car, a refrigerator, an electric light, etc.) that one intends to or is requested to permit to remain in the running state.

Both of those can also take "off" instead of "on" as a complement to turn the meaning around into a negative.

Please keep [or leave] the light off. I'm trying to sleep.

To "hang on to" something is a bit different. It can mean one intends to or is requested to not get rid of something.

Hang on to your hopes.

Hang on to that receipt. You'll need that if the IRS audits you.

"Hang on" can also be a request for someone to wait.

Hang on! I'll be there in a minute.

Note that using "off" changes the meaning into something that is not the opposite. It means to literally suspend something or someone from something else.

I wouldn't want to hang off the side of a building until we see world peace.

"Leave off" also can refer to a point where someone stopped doing something:

Where did I leave off in the story I was reading to the kids?

And "keep off" can be a warning against trespassing:

Keep off the grass!

Answer 2

To simplify the learning of English phrases, keep in mind that those little words ("prepositions" or "adverbs" or "particles" to grammarians) originally represented a very clear meaning of place or movement. The basic meaning of on is touching or in contact with. Once you understand this idea, the meaning of a phrase will usually become transparent.

On in put your jacket on means to place your jacket in contact with your body.

Keep your jacket on includes the idea of keep, "maintain, do not change" or, in other words:

Once your jacket is on you, do not remove it.

Leave carries the idea of go away from a place or thing, so leave your jacket on means:

Once your jacket is on you, your hands should 'go away' and not touch the jacket again.

Leave something on and Keep something on, therefore, are similar in that after you have put something on, you should not touch or change the way it is.

Hang on to your water bottle still has the same meaning of on, but adds the word hang meaning to hold something (with your hands touching on it) and not let it go. This is just another way of saying Keep your water bottle.

Students of English should remember that on means in contact with, so other meanings, unless they are foreign loan expressions, are an extension of this idea.