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Q. Illustrate what path does an asteroid take?
Answer:-
The asteroid will forever take the path minimizing its Lagrangian recognized as the least action principle in Lagrangian mechanics. With conservative forces there are four fundamental shapes for two different cases either the asteroid is trapped in the gravitational pull of an object or it is not trapped. If the asteroid's energy is little enough to be trapped then it can have either a circular or elliptical orbit. If it isn't trapped then this free asteroid (much like a free electron) can have either a parabolic or a hyperbolic path. A lot of this information came from Classical Mechanics by kibble and Berkshire even though there are much better books.
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