A problem Understanding how a two-body system of planets starts rotating around barycentre Consider,We are Creating a Two-body system in free space,Where no other mass exists,Let's Take First Mass M1 and hold it,Now bring Second Mass M2,hold it up,Now we are giving a suddenly impulse To M1 causing its initial Velocity to be V1 and then releasing M2,now here,How do we derive that These bodies will perform Circular motion About (Barycentre/Centre of masses),What will happen between The the time When Impulse is given and time When They become steady rotating about Centre Of mass(in COM frame).
The Gravitational force is only considered here,So my Question is Centrepetal Force Acts on both which varies inversely to $r^2$,So How are they set into circular motion?
 A: In your model, there is no expectation of circular motion.
The usual theoretical understanding is that planets form from gas and dust in an accretion disk. The gas and dust in a disk moves in nearly circular orbits: frictional effects (not considered in your model) are assumed to damp radial motion. The planets (initially) share that motion.
A: Planets and stars are formed from already rotating protoplanetary disk of dust and gas, from which planets "borrow" angular speed. Of course there's still question remains - how and when (presumably first) protoplanetary disks acquired angular momentum ? I believe root causes lies in Navier-Stokes equations and turbulence. Dust/Gas mass in the galaxy is affected randomly by local gravitational sources (many-body problem), which produces friction between layers of dust, and consequently - relative angular motion and turbulence, which fragments dust into disks.
A: The velocity $\mathbf v_1$ of $M_1$ must be such that there is not radial component when $M_2$ is released. If this condition is fulfilled, there is a specific $|\mathbf v_1|$  for a circular orbit.
A: Your question seems to indicate that you think of the rotational motion around the barycentre as some stationary state the two-body system reaches after some short time after the initial conditions are set, and only at that point is it possible to speak about circular motion.
This is not the way Newtonian mechanics works. The rotation starts as soon as the two bodies are set at their initial positions with their initial velocities. However, to have it, not all the initial conditions are good. If you put the two bodies at any distance with zero speeds or with non-zero collinear velocities, their motion will remain confined to a line segment. This is a consequence of starting with zero (conserved) angular momentum. Instead, the angular momentum is different from zero if the initial velocities are not aligned with the relative position vector. In such conditions, the motion remains confined in a plane. Still, the trajectories of the two bodies immediately start to bend as a consequence of the existence of an acceleration not aligned with the velocity.
Then, for a general attractive force between the two bodies, it is possible to show that the motion in the system frame where the center of mass is at rest will remain confined in the region between two concentric circles (annulus). In general, the trajectory will fill this region densely, but for the Newtonian forces with intensity proportional to $\frac{1}{r^2}$, the trajectory is a closed ellipse in general.
I stress that such behavior is something characterizing the motion immediately from the starting conditions. It is not some final stage of the motion.
