Question: Given a collection of point-particles, interacting through an attractive force $\sim \frac{1}{r^2}$.

Knowing only $m\_1a=\sum\_i \frac{Gm\_1m\_i}{r^2}$ and initial conditions we can deduce the motion of the system.

Consequently we can observe that three quantities remains constant
A) center of mass of the system
B) total energy
C) angular momentum

How can we derive these 3 facts directly from $m\_1a=\frac{Gm\_1m\_2}{r^2}$ ?

Are these quantities conserved for any attractive force $\sim\frac{1}{r^n}$ ?

Given any monotonically decreasing force for $r$ in $(0,\infty)$, which are the conserved quantities?

Answer: 
The total momentum is also conserved.

Noether's theorem relates conserved quantities to continuous symmetries of the system.

For the inverse square law, there's also a less known symmetry
$x \to e^\lambda x$, $t \to e^{2\lambda} t$.

But we shouldn't expect any more because of chaos; this isn't an integrable system.
All codes of the N-body simulation make use of instantaneous propagation of the speed of gravity.
The other answers follow the same criterion that is good enough for local simulations.

**As a consequence of the finite speed of propagation of gravity and the motion of the masses there is a dissipative component.**

As the bodies are never static, the masses continuously lose energy to space because there is no known mechanism that reverses the direction of flow.

(An equivalent effect applies to the radiation of charged particles by virtue of its continued accelerated motion.)

from **Jorgen Kalckar and Ole Ulfbeck experiment** (a) (energy of two masses and a spring):
...

> 
> The difference between these two
>  energies is lost by each mass: it is
>  taken away by space-time, in other
>  words, it is radiated away as
>  gravitational radiation
> 
> 
> gravity can behave like a wave:
>  gravity can radiate
>  ...
>  All this follows from the expression of universal gravity when applied to moving observers, with the requirement that neither observers
>  nor energy can move faster than c.
> 
> 
> 

(a) ***motionmountain*** free online book, ch 18, pag 526/527: Motion in General Relativity, Gravitational Waves.