The Lagrange Equations

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


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.


Lagrange Equation Virtual Work Equilibrium Configuration Virtual Displacement Total Collapse 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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