Notes on Lagrangian continuum mechanics

In Continuum Mechanic a simple material body $\mathcal{B}$ is represeted by a three-dimensional differentiable manifold and the configuration space is given by the space of embeddings $Emb(\mathcal{B},{\mathbb{R}}^n)$. We use the topology of infinite-dimensional manifold of this space, to present the first variation formula for Lagrangian mechanics.