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RE: Is there a more systematic approach to classical mechanics than Newtons second law?
Followed by Hamiltonian mechanics.
I'm not sure it would qualify as "easier", but it is more "elementary". The question asks about classical mechanics, so the next step that led to quantum mechanics is beyond the scope of the question.
I would say that Lagrangian mechanics made my life a lot easier while studying Physics because you could practically turn off your brain even in very hard multi-object systems, as everything is based on scalars and the minuses and pluses come out right by definition. So I would say it is "easier".
Seems such a long time ago... memory can't push past the haze of quantum mechanics :-)