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For variational principles in PDE see Sections 10.3 and 10.4, 10.5 from PDE-textbook.
Homogeneous isotropic elasticity system originates from expression for deformation energy \begin{gather} E:= \iiint_X \Bigl( \lambda \sum_{j,k} \sigma_{jk}^2 + \mu (\sum_j \sigma_{jj})^2\Bigr) \,dx \label{eq-5.A.1} * \end{gather} where \begin{gather} \sigma_{jk}= \frac{1}{2}(u_{j,x_k} + u_{k,x_j}) \label{eq-5.A.2} \end{gather} are components of the deformation tensor, $u_j$ are components of the displacement and $dx=dx_1dx_2dx_3$.
This leads to the variation of $E$ \begin{multline} \delta E= \int_X \sum_{j,k} \Bigl( \lambda \bigl(u_{j,x_k} + u_{k,x_j}\bigr) \delta u_{k,x_j}+ 2\mu u_{j,x_j} \delta u_{k, x_k}\Bigr)\,dx =\\ \begin{aligned} -&\int_X \sum_{j,k} \Bigl((\lambda +2\mu) u_{j, x_jx_k} + \lambda u _{k,x_jx_j}\Bigr)\delta u_k\,dx \\ -& \int_{\partial X} \sum_{j,k} \Bigl( \lambda \bigl(u_{j,x_k} + u_{k,x_j}\bigr) \nu_j+ 2\mu u_{j,x_j} \nu_k\Bigr)\delta u_{k}\,dS \end{aligned} \label{eq-5.A.3} \end{multline} where $dS$ is an area element and $\nu_j$ are components of the unit inner normal.
The first line in the right-hand expression of (\ref{eq-5.A.3}) leads to the system \begin{gather} \rho u_{k,tt}= \sum_j \Bigl(\lambda u_{k,x_jx_j} + (\lambda +2\mu) u_{j,x_jx_k}\Bigr) \label{eq-5.A.4} \end{gather} which is (5.5.1).
Consider the second line in the right-hand expression of (\ref{eq-5.A.3}).
If points on the boundary are fixed, then $\delta u_j|_Y=0$ this line is $0$, so we have all boundary conditions $u_j|_Y=0$.
If points on the boundary can slide along boundary freely but not in the normal direction, then this expression must vanish as $\sum_j \nu_j \delta u_j=0$, that is \begin{gather} \sum_j (u_{j,x_k} +u_{k,x_j})\nu_j|_Y = p \nu_k, \label{eq-5.A.5} \end{gather} which is added to \begin{gather} f(x_1+u_1,\ldots, x_3+u_3)|_Y=0 \label{eq-5.A.6} \end{gather} where $f(x_1,x_2,x_3)=0$ is an equation of $Y$. Excluding $p$ from (\ref{eq-5.A.5}) we get three boundary conditions.
If boundary is free then $\delta u_k$ on $Y$ are arbitrary and we have a boundary conditions \begin{gather} \sum_j \bigl(\lambda (u_{j,x_k}+u_{k,x_j}) \nu_j +2\mu u_{j,x_j}\nu_k\bigr)\bigr|_Y=0. \label{eq-5.A.7} \end{gather}