So not that we know how to solve differential equations in principle or in a single dimension.
We can now turn to the fun part.
We can describe an object in multiple dimensions or even several objects at the same time.
So we will first learn how to do this.
And I can tell you, it's really not that difficult when we know how the background works and then we
can solve several examples, but we will start with a single object in several dimensions.
So the first example will be propagating a rolling ball, and then we will consider a so-called Lorence
system, which can give rise to chaotic solutions.
We will learn about the butterfly effect and then we will discuss multiple objects at the same time
on multiple points of a sample.
So the first example of this will be solving the heat equation.
So we take a material and heated on one side, and then we investigate how the temperature or how the
heat propagates through the system over time, so we can then calculate at every point of the sample
at any time the temperature.
So that's pretty difficult, but we can do it with our methods that we have just developed.
And then we turn to really a fun example, which is the three body problem.
So we will first solve the propagation of the Sun, the Earth and the Moon, and we will use here really
the actual parameters of real life.
And we can really figure out why the Earth orbits the sun and in 365 days, and also why the Moon orbits
the Earth.
And then we can really plan for a trip to the Moon and even beyond.
So we will add here a fourth object, which is a spaceship with an astronaut, and we can then model
different trips.
So just by tuning the starting velocity, we will, for example, achieve an orbiting motion around
the Moon.
Or we can even exit the Earth's gravitational influence, and then our spaceship will orbit the Sun
on its own.
So you can really look into the future, and you should try to plan some trips of future spacecraft
missions.