The Lagrange Equations

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.