Classical Mechanics pp 141-172 | Cite as

# 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.