If a particle starts at a point and travels in a circular path and comes to it's initial position, is the energy wasted.The particle should have the same energy at the end.
If so what about the energy required for travelling?
If a particle starts at a point and travels in a circular path and comes to it's initial position, is the energy wasted.The particle should have the same energy at the end.
If so what about the energy required for travelling?
No energy is required for traveling. A particle in motion has kinetic energy, but this is regardless of the journey and only about the speed.
You are right that when the particle stops, something must have stopped it. That something has then absorbed the kinetic (motion) energy.
So the main answer is: No! Energy cannot be destroyed.
Energy can neither be created nor be destroyed. You are only converting one form of energy to another in the example you gave.
If a particle travels in a loop, it will need to change its velocity. This requires force, which requires work to be done.