Newtonian mechanics – Which classical physics situations are not determined?

There are two famous cases of mechanical mechanics that fail to be determined.

The first, and most famous, is Norton’s dome, equal to a system with a strong form

$$ f = \ sqrt {r} $

There are many details of Wikipedia Article (This is usually described as a result of a reaction force from a surface with a specified form), but the main idea is that the cause of energy fails to explain $ r = 0 $, since

$$ (\ sqrt {r}) ‘= \ frac {1} {2 sqrt {r}}

Consequently, no guarantee that the equation is $ \ ddot {r} = \ sqrt {r} this is a unique solution (and it is not it), because it is not Continue Lipschitz.

There are many information about Norton’s dome, here and on the Internet, so the more interesting, if more pathology, for example, the invader of space.

Space Invader is a particle submitted by an unstoppable acceleration at the last hour, so that it comes “Infinity” after. Exactly form of energy does not matter, but for example you can choose

$$ f = \ tan

In such cases, the fragment that goes into infinity to $ t = \ pi / 2 $ and, after time, stop. As this system is time-symmetric, it is also possible to consider the case with a fragment that is initially do not and comes from the emptiness of the consequences of results).

Another example of such behavior is Paint diseases without a collision. The most famous example of which is a 5-body gravity problem in which one of the parties also goes infinity at the end of the last season, only by borrowing energy from two 2-body systems. As for point-particles, potential energy is not prevented from below (because it is $ e \ Propto -1 / R $), it is possible to have an endless kinetic force while continuing to Be careful of energy, by having the 2-body system with its collapse.

For a general treatment of the issue of determining physical physics, you can also check This Earman articlefor example.