Given a variational problem
$$ J[x(t)]=\int_{\Omega}L(t,x(t),\frac{dx}{dt}(t))dt $$a Noether symmetry (also known as divergence symmetry) is a one-parameter local group of transformations
$$ t'=t'(t,x,s)\quad \quad x'=x'(t,x,s) $$such that
$$ L\left(t',x',\dot{x}'\right)dt'=L\left(t,x,\dot{x}\right)dt+dF(t,x,s) $$for all values of $s$ where the transformation is defined and for some smooth function $F$. When $F=0$ it is called a variational symmetry.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: