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Newtonian Mechanics
Think of particles in Newtonian physics interacting, let's say, through gravity. Or even forget gravity. Let's say they interact as billiard balls. So a big box with bouncy walls with lots of billiard balls bumping into each other. You have n points in three dimensional space and you actually also need the momenta or the velocity of the particles. In total, you have six n dimensions to completely describe the state of such a system.