Noether symmetry

J [x (t)] = \int_{Ω} L (t, x (t), \frac{d x}{d t} (t)) d t

a Noether symmetry (also known as divergence symmetry) is a one-parameter local group of transformations

t^{'} = t^{'} (t, x, s) x^{'} = x^{'} (t, x, s)

such that

L (t^{'}, x^{'}, {\dot{x}}^{'}) d t^{'} = L (t, x, \dot{x}) d t + d F (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.