In this chapter, we explain the key points in the framework of analytical mechanics. This section can be regarded as the “core of the book” and serves as the “trailhead” for beginners venturing into quantum physics. Some readers find it easier to understand quantum mechanics using an axiomatic or formal scientific style, namely, understanding it “from the rules”. However, there are others who find it more intuitive to trace “how the ideas were originally developed” (Sect. 1.4.1 ). For the latter group, introducing wave mechanics through the optical-mechanical analogy from the Hamilton-Jacobi formulation of analytical mechanics provides the “most seamless trail” for the ascent. The content of this chapter provides the minimum path necessary to reach the Hamilton-Jacobi formulation. Furthermore, the material covered up to Chap. 2 can be considered the “pre-hike preparations” required before tackling this chapter.