I'm looking for a more general explanation, not specific to a single case. There are many situations where energy has to be expended to hold something up - a hovering helicopter, lifting with your arm, using an air jet to simulate skydiving - and so on. In all of these cases, an object is being held in a static position (like a book on a table), but energy has to constantly be expended to maintain it (unlike the book). Why is this?
A hovering helicopter is falling. In order to maintain it's position, it must produce a downdraft - spending energy to force air downward to lift itself up. It is exerting a force on the air. A book on the floor requires energy to be spent to lift it to the table. In both cases the same amount of energy is expended: An amount equal to the amount of potential energy gained by the object. The difference is, once the book has been moved, it stays there. No more energy is being spent by it or the table to hold it up, despite the fact that the book is exerting a force on the table - just like the helicopter is on the air. Except that the helicopter has to keep spending energy to exert that force, and the book doesn't.
Energy is being transferred from the helicopter (chemical, in the fuel) into the air (as motion). The helicopter is, net, losing energy in order to maintain it's position.
Why can force sometimes be exerted for free (the table has a constant force on the book), but sometimes has a cost?