I'd like to shutdown Arduino as much as possible, in order to save power.
So, I would like to start the arduino with mechanical action/trigger (pressure button, magnetic sensor for door, ...)
I want it to stop by itself once it have done what it have to do.
So do you have any exemple or any feasibility issue to report?
if you use a relay, you can have the button close the relay and apply power to the arduino.
the arduino would then run.
the arduino could open a second relay and kill power to itself.
you could use mosfets that back-fed itself to stay on with a button press
and the use the arduino to remove power.
this is very simple stuff, so you should be able to make it work without too much trouble.
I would highly suggest you tell us what your device would do.
why you want too save power
there is a thing called an X/Y problem in which the ideas of the student are not aware of the vast array of options so pics what seems like the only possible choices. and are usually made after multiple previous mis-understandings of what can be done.
the result is a work-around around a work-around that will waste time and not really work well.
an arduino is an energy hog in the 'off the shelf' version, but a few simple adjustments can make it a very energy efficient device.
you could waste more power in running a stock arduino for an hour than letting it sleep for a week.