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The Lagrange Equations

  • Emmanuele DiBenedetto
Chapter
Part of the Cornerstones book series (COR)

Abstract

Let \(\{\mathcal{M};d\mu \}\)be a material system whose mechanical state is described by N Lagrangian coordinates \(q = ({q}_{1},\ldots, {q}_{N})\). Since every point \(P \in \{\mathcal{M};d\mu \}\)is identified along its motion by the map (q, t) → P(q, t), the configuration of the system is determined, instant by instant, by the map \(t \rightarrow q(t) : \mathbb{R} \rightarrow {\mathbb{R}}^{N}\). The latter can be regarded as the motion of some abstract point in some N-dimensional space, called configuration space. Since N is the least numberof parameters needed to identify uniquely the position of each point P of the system, each of the maps \(\{\mathcal{M};d\mu \} \ni P \rightarrow \| \partial P/\partial {q}_{h}\|\), \(h = 1,\ldots, N\), is not identically zero. Equivalently, we have the following lemma.

Keywords

Lagrange Equation Virtual Work Equilibrium Configuration Virtual Displacement Total Collapse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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