Simple harmonic motion

and

Our particle obeys Newton's second law of motion,

(2.58) |

Let us assume that the particle always stays fairly close to its equilibrium point. In this case, to a good approximation, we can represent via a truncated Taylor expansion about this point. In other words,

(2.59) |

However, according to Equations (2.56) and (2.57), the preceding expression can be written

where . Hence, we conclude that our particle satisfies the following approximate equation of motion,

provided that it does not stray too far from its equilibrium point; in other words, provided does not become too large.

Equation (2.61) is called the *simple harmonic equation*; it
governs the motion of all one-dimensional conservative systems that are slightly
perturbed from some stable equilibrium state. The solution of Equation (2.61)
is well known:

The pattern of motion described by this expression, which is called

Note that the frequency, --and, hence, the period, --of simple harmonic motion is determined by the parameters appearing in the simple harmonic equation, Equation (2.61). However, the amplitude, , and the phase angle, , are the two integration constants of this second-order ordinary differential equation, and are thus determined by the initial conditions; that is, the particle's initial displacement and velocity.

From Equations (2.45) and (2.60), the potential energy of our particle at position is approximately

(2.63) |

Hence, the total energy is written

(2.64) |

giving

where use has been made of Equation (2.62), and the trigonometric identity . Note that the total energy is constant in time, as is to be expected for a conservative system, and is proportional to the amplitude squared of the motion.

Consider the motion of a point particle of mass that is slightly displaced from a unstable equilibrium point at . The fact that the equilibrium is unstable implies that

(2.66) |

and

(2.67) |

As long as remains small, our particle's equation of motion takes the approximate form

(2.68) |

which reduces to

(2.69) |

where . The most general solution to the preceding equation is

(2.70) |

where and are arbitrary constants. Thus, unless the initial conditions are such that is exactly zero, the particle's displacement from the unstable equilibrium point grows exponentially in time.