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 $Emb \left( \mathcal{B} , \mathbb{R}^{n} \right)$. We use the topology of infinite-dimensional manifold of this space, to present the first variation formula for Lagrangian mechanics.