Introduction to Analytical Dynamics pp 147-160 | Cite as

# Oscillations

Chapter

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## Abstract

In this chapter, we look at an analogue of simple harmonic motion in systems with many degrees of freedom. This is not only of interest in itself, but it also gives a good approximation to the behaviour near equilibrium of a general class of mechanical systems.

With just one degree of freedom, one can characterize simple harmonic motion by the equation of motion or equivalently by the form of the Lagrangian that generates the motion. The constant ω is the angular frequency of the oscillations. Angular frequency is measured in radians per second and is related to frequency ν, which is measured in hertz (Hz), or cycles per second, by ω = 2πν.

$$\ddot{q} + {\omega }^{2}q = 0,$$

$$L = \frac{1} {2}{v}^{2} -\frac{1} {2}{\omega }^{2}{q}^{2}\,$$

## Keywords

Analytical Dynamics Normal Mode Characteristic Equation Angular Frequency Fundamental Solution
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© Springer-Verlag London 2009