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So not that we know how to solve differential equations in principle or in a single dimension.

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We can now turn to the fun part.

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We can describe an object in multiple dimensions or even several objects at the same time.

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So we will first learn how to do this.

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And I can tell you, it's really not that difficult when we know how the background works and then we

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can solve several examples, but we will start with a single object in several dimensions.

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So the first example will be propagating a rolling ball, and then we will consider a so-called Lorence

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system, which can give rise to chaotic solutions.

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We will learn about the butterfly effect and then we will discuss multiple objects at the same time

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on multiple points of a sample.

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So the first example of this will be solving the heat equation.

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So we take a material and heated on one side, and then we investigate how the temperature or how the

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heat propagates through the system over time, so we can then calculate at every point of the sample

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at any time the temperature.

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So that's pretty difficult, but we can do it with our methods that we have just developed.

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And then we turn to really a fun example, which is the three body problem.

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So we will first solve the propagation of the Sun, the Earth and the Moon, and we will use here really

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the actual parameters of real life.

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And we can really figure out why the Earth orbits the sun and in 365 days, and also why the Moon orbits

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the Earth.

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And then we can really plan for a trip to the Moon and even beyond.

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So we will add here a fourth object, which is a spaceship with an astronaut, and we can then model

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different trips.

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So just by tuning the starting velocity, we will, for example, achieve an orbiting motion around

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the Moon.

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Or we can even exit the Earth's gravitational influence, and then our spaceship will orbit the Sun

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on its own.

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So you can really look into the future, and you should try to plan some trips of future spacecraft

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missions.