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In Newtonian mechanics we all learned that a mass accelerates when it is under the influence of a nonzero net force.

However, I learned of D'Alembert's Principle today and it seems to oppose what I learned in Newtonian mechanics. For example, what we (or at least I) considered to be fictitious forces (eg. inertia) is now suddenly relevant. Furthermore, the applied and inertial "forces" are equal when the object is accelerating!

I am having difficulty in getting an intuitive understanding of how D'Alembert's Principle works and why it doesn't contradict Newton's laws.

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Well, fictitious forces are also relevant for Newtonian mechanics in accelerated reference frames (see e.g. xkcd/123/ for a humorous take), so in that sense d'Alembert's principle is no different. At its core d'Alembert's principle is expressing Newton's 2nd law, cf. this related Phys.SE post and links therein.

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