Gravitational N-body problem: direct summation | 0:38: Visually useful, but actually wrong about the number of gravitational force calculations needed using Newton's law of universal gravitation. The video calculates each interaction explicitly: 10 point particles, 9 interactions each, so 10*9 = 81 calculations. This is inefficient. To explicate: Newton's law of universal gravitation explicitly obeys Newton's 3rd law of motion: for every force there is an equal (in magnitude) and opposite (in direction) force. So you actually only need to calculate one of the pair of gravitational forces between two point particles. The total number of calculations M for N point particles is given by:
```
  M = sum_(k=1)^(N) (N-k) = N**2 - sum_(k=1)^(N) k = N**2 - N(N+1)/2 = N(N+1)/2 
```
(see Wikipedia: Summation: Powers and logarithm of arithmetic progressions). The number of calculations for N = 10 is thus M = 55, and NOT 81.
Having solved for the gravitational forces, one can calculate the motions of the point particles over a short time step and one then advances their positions accordingly. Then one recalculates their gravitational forces and so on. The procedure is the essence of N-body simulations.
Short enough for classroom.
N body simulation in Python with code (precision approach) | 1:01: Good, but is this full 3-dimensional and shown in 2-d projection or are the particles restricted to 2-dimensional space. It's hard to tell, but yours truly believes the N-body simulations are artificially confined to 2-dimensional space. The N-body simulations for N = 3, 4, 13, and 30 point particles calculated using a Python code. Full information is NOT given with the video, but yours truly assumes the particles have equal mass and are point particles (i.e., they have zero size, and so NEVER collide in a body-body sense NOR merge). The initial conditions must chosen in some good fashion for illustrative purposes. Yours truly believes that initial conditions are such the system is gravitationally bound which is equivalent to saying kinetic energy plus gravitational potential energy sums to less than zero. Being gravitationally bound means that NOT all the point particles can go to infinity relative to each other. Due to complex gravitational interactions (including explicit gravity assists (AKA gravitational slingshot maneuvers) can eject individual point particles to infinity, but the remaining ones are more tightly gravitationally bound. In some cases depending on initial conditions, so many ejections may occur that only 2 point particles, and they can NEVER escape from each other.
      Note, there is NO exact general analytic solution for 3 or more bodies interacting via gravity (see Wikipedia: Three-body problem). Usually, one must solve such systems by perturbation theory or numerical methods (e.g., N-body simulations) on the computer. There is an exact general analytic solution for gravitationally bound 2-body system within Newtonian physics.
      Note also, open star clusters if they are formed bound (and they NOT be) often suffer considerable escape during their existence to internal and external gravitational interactions and disperse in a few millions to tens of millions of years, a process often called "evaporation" (see Wikipedia: Open cluster: Eventual fate).
      Short enough for classroom.
N-body simulation of 400 objects with parameters close to those of Solar System planets | 0:57: This video of an N-body simulation is somewhat similar to the one just above, but astro-bodies have nonzero size and can merge. If the time clock is real, yours truly believes the astro-bodies are probably small planetesimals. For the Solar System, there should be an overall rotation, but it seems likely to yours truly that the N-body simulation is using periodic boundary conditions in a simulation box. Short enough for classroom.
N-body galaxy simulation 1 | 1:30: An N-body simulation of the formation of a galaxy using 10,000 point particles presumably of equal mass. However, the caption does NOT state if these point particles are meant to represent baryonic matter starting in a disk with some reasonable initial conditions or that AND and an invisible dark matter halo, roughly spherically symmetric and containing ∼ 10 more mass than the disk. Yours truly guesses the former case which is less realistic, but computationally easier: the N-body simulation is just an example calculation after all. There are, of course, NO stars, star formation, or interstellar medium (ISM). The point particles just represent mass interacting via gravity: this is an N-body simulation recall. The disk is ∼ 30 kpc and the calculation runs for 0.380 gigayears. Why does the galaxy become a barred spiral? A choice in the initial conditions? Will it remain a barred spiral or evolve, maybe chaotically? Short enough for classroom.

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