Why is the Kinetic energy of the system minimum in the frame of the center of mass for any number of particles
solution) The kinetic energy of any entity depends on the reference frame in which it is measured. Though the total energy of an isolated system, i.e. one which energy can neither enter nor leave, does not change in whatever reference frame it is calculated. Therefore, the chemical energy converted to kinetic energy by a rocket engine is divided differently among the rocket ship and its exhaust stream depending upon the chosen reference frame. This is known as the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with dissimilar reference frames disagree on the value of this conserved energy.
The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.