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Analytical Mechanics for Relativity and Quantum Mechanics$
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Oliver Johns

Print publication date: 2011

Print ISBN-13: 9780191001628

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780191001628.001.0001

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Small Vibrations about Equilibrium

Small Vibrations about Equilibrium

(p.242) 10 Small Vibrations about Equilibrium
Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Davis Johns

Oxford University Press

This chapter focuses on the one or more essentially stable equilibrium configurations of mechanical systems. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. The Lagrangian theory of these small vibrations is presented here for the simple case of systems with a finite number of degrees of freedom. The chapter defines equilibrium by using the example of a marble. A marble placed at rest at the bottom of a spherical bowl will remain there forever. A marble placed at rest, and very carefully, on the top of a sphere will also remain there so long as no forces other than gravity act, as will a marble placed on a flat, level tabletop. These examples illustrate the three types of equilibrium point. The first is called stable; the second, unstable; and the third, conditional.

Keywords:   equilibrium configurations, mechanical systems, normal modes, Lagrangian theory, degrees of freedom

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