Exercise 1

A closed system can be considered independent of its environment. There are no interactions with the outside world that could influence the state or behavior of the system. There is no exchange of particles or energy with the outside world, i.e. the number of particles and the energy must be conserved within the closed system (conservation laws).

Strictly speaking, such a system does not exist. However, in order to keep the number of variables manageable, one is forced to consider multiple isolated systems, otherwise the system as a whole would be far too complex. Closed systems are always an approximation, but very often this approximation is very good and sufficiently accurate.

Example: If you look at the movement of an electron in a magnetic field, the influence of the earth’s gravity can be neglected and of course also the influence of the gravitational forces of other planets and the sun, although gravitational forces cannot be shielded and have an infinite range. The influence of these forces on the motion of the electron is so small that it plays no role in the accuracy of measurement and calculation.

However, it should also be mentioned that the influence of the initial conditions in complicated mathematical models can be very high. If the initial conditions are changed even slightly, the calculated final state can be completely different. This is where you reach the limits of calculability with mathematical models, because the initial conditions of dynamic systems are not exactly known. Examples of this are the weather forecast over a longer period of time (butterfly effect) or the state of billiard balls on the table after a large number of shots.

In classical mechanics, an example of such a system is the double pendulum (s. Exercise 6, Lecture 7).