A Low Temperature Stirling Engine is a closed cycle heat engine.
To understand how a Stirling Engine works, we need to identify its major components. Take a look at the picture below for a run down on the major parts.
The Stirling Engine requires a temperature difference between two plates in order to run. In this case the plates are made from aluminium. This is a a good choice because aluminium is highly conductive.
In order to power the Stirling Engine, we need to have one plate hotter than the other. This is easily achieved by placing the Stirling Engine on a cup of hot water.
The gap between the two plates is sealed, containing a fixed volume of air. As the bottom plate warms up, the air between the plates expands, pushing up the piston that seals a hole in the top plate. This piston is attached to the flywheel by a metal rod.
As the piston rises, the flywheel is turned. The movement of the flywheel in turn pushes down a second rod which is attached to a displacer sitting loosely between the plates. As the displacer moves downwards, it pushes air away from the hot bottom plate and up to the top plate, which is cooler. This causes the air to contract and the piston is pulled back down again, turning the flywheel further.
The turning flywheel raises the displacer again, pushing air back to the hot plate, and so the cycle continues while there is still a sufficient temperature difference between the two plates.
Because the Stirling Engine needs a temperature difference between the top and bottom plate to run, you don't need a heat source to power it. Ice can work just as effectively. In the video below you can see a Stirling Engine running on dry ice. The bottom plate is kept cooler than the top. The air in contact with the bottom plate cools and contracts, pulling the piston down which turns the flywheel. The flywheel pushes the displacer down, moving air away from the cool plate to the warmer upper plate. The air warms and expands, pushing the piston back up.
As you can see, it isn't the input of heat that is important to running the Stirling Engine, but rather the creation of a significant temperature difference between the two plates. Eventually the temperature difference between the two plates will drop. Once the plates are nearly the same temperature, the Stirling Engine will no longer run.
Stirling Engines do not require a flywheel to run, however at low temperatures differentials it is much easier to keep a stirling engine with a flywheel running as the momentum from the flywheel will keep the displacer moving.
In the example Free Piston Stirling Engine shown in the video below, the mechanism works as follows:
The piston is lifted up and the displacer is down. The piston is dropped and hits a spring, moving the displacer up. This moves the cold air to the bottom of the chamber, where it is heated by the bottom plate. Air expands, pushing the piston back up as the displacer falls back down under gravity, moving the air from the bottom to the top plate. Air cools and contracts, pulling the piston back down. The piston hits bottom spring, moving the displacer back up.
Where a greater temperature difference is present, even a very small scale Stirling Engine can generate a significant amount of kinetic energy. See the video below of a small Flame Powered Stirling Engine.
Because of the need for a snug fit of the piston to make an airtight seal, building a Stirling Engine from scratch can be tricky. However kits are available which allow you to construct your own Stirling Engine while forgoing the need for machining parts. The videos below show two examples of Low Temperature Stirling Engine Kits running on a cup of hot water.