Definition 4.10. @olver86
A local group of transformations $G$ acting on $M \subseteq \Omega_0 \times U$ is a variational symmetry group of the a variational problem $J[u]=\int_{\Omega}L(x,u^{(n)})dx$ if whenever $\Omega$ is a subdomain with closure $\bar{\Omega} \subseteq \Omega_0$, $u = f(\textbf{x})$ is a smooth function defined over $\Omega$ whose graph lies in $M$, and $g \in G$ is such that $\tilde{u} = \tilde{f}(\textbf{x}) = g \cdot f(\textbf{x})$ is a single-valued function defined over $\Omega \subseteq \Omega_0$, then
$$ \int_{\hat{\Omega}} L(\textbf{x}, \text{pr}^{(n)}(\tilde{f}(\textbf{x}))) d\textbf{x} = \int_{\Omega} L(\textbf{x}, \text{pr}^{(n)}(f(\textbf{x}))) d\textbf{x}. $$$\blacksquare$
We are going to proceed with one independent variable $t$ and one dependent variable $x$ for simplicity, but can be generalized to any bundle $E$ with coordinates $(x_i,u^{\alpha})$.
A variational symmetry group (even a Noether symmetry group) sends solutions of the Euler-Lagrange equations associated to $L$ to solutions. Observe that:
$$ \begin{aligned} \int_{t_{0}^{\prime}}^{t_{1}^{\prime}} L\left(x^{\prime}, \frac{\mathrm{d} x^{\prime}}{\mathrm{d} t^{\prime}}, t^{\prime}\right) \mathrm{d} t^{\prime} &=\int_{t_{0}}^{t_{1}} L\left(x, \frac{\mathrm{d} x}{\mathrm{~d} t}, t\right) \mathrm{d} t \\ &+\int_{t_{0}}^{t_{1}} \frac{\mathrm{d}}{\mathrm{d} t} F(x, t, s) \mathrm{d} t \end{aligned} $$and since the last term in the right hand side is constant for curves with the same endpoints, a curve maximizes/minimizes the lhs if and only if maximizes/minimizes the firs term of the rhs. This is formalized for variational symmetries in Theorem 4.14 in @olver86. In particular, if $G$ is a variational symmetry group of the variational problem $J$ then it is a symmetry group of the associated Euler-Lagrange equations.
I suppose that Noether symmetries and variational symmetries give rise to a distinguished kind of generalized symmetries, even of Lie point symmetrys, of the corresponding Euler-Lagrange equations.
Theorem 4.12. @olver86.
Theorem
A connected group of transformations $G$ acting on $M \subseteq \Omega_0 \times U$ is a variational symmetry group of the functional $J$ if and only if
$$ \text{pr}^{(n)} v(L) + L \text{Div} \xi = 0 $$for all $(x, u^{(n)}) \in M^{(n)}$ and every infinitesimal generator
$$ v = \sum_{i=1}^{p} \xi^i(x, u) \frac{\partial}{\partial x^i} + \sum_{\alpha=1}^{q} \phi_\alpha(x, u) \frac{\partial}{\partial u^\alpha} $$(note that $J$ above is "of first-order" but it can be generalized straightforwardly).
$\blacksquare$
They are more general. See Noether symmetry.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: