Has someone an idea how to solve the 1. question for ANY number of mass points and a clue to the 2. question?

center of mass system, momentum

  • 1
    $\begingroup$ Didn't you tell me to do it like this in main question?? $\endgroup$ – Tom Jan 14 '19 at 9:24
  • $\begingroup$ Yes, but I didn't mean something like the above which just references the previous question. I meant a fully written out question with the full problem statement so the new question can stand alone. $\endgroup$ – m_goldberg Jan 14 '19 at 9:34
  • $\begingroup$ @Tom Indeed, a bit more self-containedness of this question would be desirable. When I read it the first time I thought "What the ... do they mean with '1. question'?" $\endgroup$ – Henrik Schumacher Jan 14 '19 at 9:38
  • $\begingroup$ ok, in future I will try to do it more correctly! thank you for patience ! $\endgroup$ – Tom Jan 14 '19 at 9:42
  • $\begingroup$ Near duplicate: mathematica.stackexchange.com/questions/189362/… $\endgroup$ – David G. Stork Jan 14 '19 at 18:43

The data for many particles are best stored in arrays.

n = 10000;
masses = RandomReal[{0.1, 1.}, {n}];
positions = RandomReal[{-1, 1}, {n, 3}];
velocities = RandomReal[{-1, 1}, {n, 3}];

Now, one obtains

centerofmass = masses.positions;
momenta = masses velocities;
totalmomentum = masses.velocities;
angularmomenta = MapThread[Cross, {positions, momenta}];
totalangularmomentum = Total[angularmomenta];

Performance Tuning

There is an issue with Cross: For some reason, it is way slower than it should be. Here is a compiled, listable, and parallelized version of it for 3-vectors:

cross = With[{
   code = Cross[
     Table[Compile`GetElement[X, i], {i, 1, 3}],
     Table[Compile`GetElement[Y, i], {i, 1, 3}]
  Compile[{{X, _Real, 1}, {Y, _Real, 1}},
   CompilationTarget -> "C",
   RuntimeAttributes -> {Listable},
   Parallelization -> True,
   RuntimeOptions -> "Speed"

It is about 500 times faster than Cross on my machine and it is already Listable so that we don't need MapThread:

angularmomenta2 = cross[positions, momenta];

Up to machine precision, the result is the same:

Max[Abs[angularmomenta - angularmomenta2]]


  • $\begingroup$ you´re genious Henrik! thank you! $\endgroup$ – Tom Jan 14 '19 at 9:43
  • $\begingroup$ Huh, thanks for the flowers. You're welcome. $\endgroup$ – Henrik Schumacher Jan 14 '19 at 9:43

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