For a system being actuated by a motor, the force can be amplified by gearing. The energy is being used for force instead of distance, so it produces more torque but moves slower.
For a system being actuated with hydraulics, the force can be amplified by having a larger cross-sectional area for the compression piston than the load piston. The small area of the compressor piston means high pressure, then this can be multiplied by the area of the large piston creating larger forces. The key to the conservation of energy is that the volume of fluid in and out the pistons is the same, so if the load piston has a larger cross sectional area it will move up less. Energy is again being used for force instead of distance.
For a system being actuated by an electro magnet (a magnetized material allowed to move inside a coil with DC current). I.e a solenoid. The force can be amplified by increasing the number of windings or by wrapping the coil around an iron tube.
In both cases (increasing windings or adding iron) how is energy conserved? I've been trying to figure this out for ages and think that the only way I will understand it is if it can be related to the mechanical leverage principles of the first two force amplification examples I gave. In both of these force is being increased at the expense of distance, hence conserving energy. When you amplify the force in a solenoid using more windings or adding iron, what is this at the expense at? i.e. how is energy conserved?