Variational symmetry

Definition

Definition 4.10. @olver86
A local group of transformations $G$ acting on $M \subseteq Ω_{0} \times U$ is a variational symmetry group of the a variational problem $J [u] = \int_{Ω} L (x, u^{(n)}) d x$ if whenever $Ω$ is a subdomain with closure $\bar{Ω} \subseteq Ω_{0}$ , $u = f (x)$ is a smooth function defined over $Ω$ whose graph lies in $M$ , and $g \in G$ is such that $\tilde{u} = \tilde{f} (\tilde{x}) = g \cdot f (\tilde{x})$ is a single-valued function defined over $Ω \subseteq Ω_{0}$ , then

\int_{\hat{Ω}} L (\tilde{x}, {pr}^{(n)} (\tilde{f} (x))) d x = \int_{Ω} L (x, {pr}^{(n)} (f (x))) d x .

$◼$

Induced symmetry in Euler-Lagrange equations

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 $(x_{i}, u^{α})$ .
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}^{'}}^{t_{1}^{'}} L (x^{'}, \frac{d x^{'}}{d t^{'}}, t^{'}) d t^{'} & = \int_{t_{0}}^{t_{1}} L (x, \frac{d x}{d t}, t) d t \\ + \int_{t_{0}}^{t_{1}} \frac{d}{d t} F (x, t, s) 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.

Infinitesimal criterion for variational symmetries

Theorem 4.12. @olver86.
Theorem
A connected group of transformations $G$ acting on $M \subseteq Ω_{0} \times U$ is a variational symmetry group of the functional $J$ if and only if

{pr}^{(n)} v (L) + L Div ξ = 0

for all $(x, u^{(n)}) \in M^{(n)}$ and every infinitesimal generator

v = \sum_{i = 1}^{p} ξ^{i} (x, u) \frac{\partial}{\partial x^{i}} + \sum_{α = 1}^{q} ϕ_{α} (x, u) \frac{\partial}{\partial u^{α}}

(note that $J$ above is "of first-order" but it can be generalized straightforwardly). Here, $Div ξ$ denotes the total divergence of the $p$ -tuple $ξ = (ξ_{1}, \dots, ξ_{p})$ . $◼$

Noether symmetry

They are more general. See Noether symmetry.