Will hydraulics lever ever stop if we exert some force to it? Suppose there is a simple hydraulics lever (like the one often shown when discussing about Pascal's Principle). If we exert some force into one of the piston without having any weight on the other piston, will the lever ever stop? I am guessing the lever will stop when the liquid pressure adjacent to the forced piston is the same as the pressure of liquid at the other side of the lever. Am I right?
 A: It depends in what you mean with "there is no weight on the other piston" . 
If you mean that there isn't an object on the other  piston but you're performing the experiment in an atmosphere then the piston will stop when there is equilibrium of forces between  the force you're exerting and the force the atmosphere exerts with its weight (note that i said equilibrium of forces and not equilibrium of pressures which would suffice it the pistons have the same surface).
If you don't have an atmosphere pushing the piston down many things can happen depending on how you push. Let's analyse a simple and nice scenario: you exert a force for a small amount of time, that is you push and the then leave piston $1$, piston $2$ is then pushed up with the fluid until they (piston $2$ and fluid) reach the maximum height and start falling, while they're coming down they push piston $1$ upwards. Then it will eventually come down push piston $2$ up and so on. So you have an armonic motion that goes on forever if the sides of the walls on which the pistons slide are frictionless and the fluid is a perfect fluid.
