In this section, we will consider two very simple analytically solvable problems, namely, the free particle and the harmonic oscillator. The former is worked out just to show that Newton's first law is a consequence of Newton's second law. The latter is studied because harmonic oscillators are a fundamental component of many classical idealized models of real systems. Harmonic oscillators are often used to describe chemical bonds, thermal motion of solids, and are actually needed in certain quantum mechanical problems that can be mapped onto equivalent classical problems (discussed in more detail in Statistical Mechanics). In both cases, we will restrict our focus to a single particle of mass, m, moving in one dimension described by a coordinate, x, and a velocity, v, for which Newton's second law reduces to
