• Nicholas M. J. WoodhouseEmail author
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


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
$$\ddot{q} + {\omega }^{2}q = 0,$$
or equivalently by the form of the Lagrangian
$$L = \frac{1} {2}{v}^{2} -\frac{1} {2}{\omega }^{2}{q}^{2}\,$$
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.


Analytical Dynamics Normal Mode Characteristic Equation Angular Frequency Fundamental Solution 
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Copyright information

© Springer-Verlag London 2009

Authors and Affiliations

  1. 1.Mathematical Institute University of OxfordOxfordUnited Kingdom

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