'phase space' of the system. Each possible state can be thought of as a point in that phase space.

As time passes, the state changes, so this representative point traces out a curve, the trajectory of the system. The rule that determines the successive steps in the trajectory is the dynamic of the system. In most areas of physics, the dynamic is completely determined, once and for all, but we can extend, this terminology to cases where the rule involves possible choices. A good example is a game. Now the phase space is the space of possible positions, the dynamic is the rules of the game and a trajectory is a legal sequence of moves by the players.

The formal setting and terminology for phase spaces is not as important, for us, as the viewpoint that they encourage. For example, you might wonder why the surface of a pool of water, in the absence of' wind or other disturbances, is flat. It just sits there, flat; it isn't even doing anything.

But you start to make progress immediately if you ask the question 'what would happen if it wasn't flat?' For instance, why can't the water be piled up into a hump in the middle of the pond?

Well, imagine that it was. Imagine that you can control the position of every molecule of water, and that you pile it up in this way, miraculously keeping every molecule just where you've placed it. Then, you let go. What would happen? The heap of water would collapse, and waves would slosh across the pool until everything settled down to that nice, flat surface that we've learned to expect. Again, suppose you arranged the water so that there was a big dip in the middle. Then as soon as you let go, water would move in from the sides to fill the dip.

Mathematically, this idea can be formalised in terms of the space of all possible shapes for the water's surface. 'Possible' here doesn't mean physically possible: the only shape you'll ever see in the real world, barring disturbances, is a flat surface. 'Possible' means 'conceptually possible'. So we can set up this space of all possible shapes for the surface as a simple mathematical construct, and this is the phase space for the problem. Each 'point' -location -in phase space represents a conceivable shape for the surface. Just one of those points, one state, represents 'flat'.