We continued exploring candidate symmetry transformations, adding
space rotations and Galilean boosts to our first two (space and time
translations). These four transformations together form the "Galilean
group", the classical analog to the Lorentz group in special
relativity. They leave the E-L equations unchanged if they change the
Lagrangian only by a total time derivative dF/dt -- a question we
investigate by direct calculation of 
.
We then derived one of the most elegant and widely valid theorems in
physics, Noether's theorem. It says that for every continuous symmetry
a system has, there exists a conserved charge Q. Specifically, for a
symmetry transformation 
, which
changes the Lagrangian by 
, the charge 
is conserved.
We then looked at the consequences of Noether's theorem for a system
of coupled oscillators whose Lagrangian was space-translation
invariant. We found that the conserved quantity given by Noether's
theorem for the symmetry of translation by 
was the
- component of linear momentum.
--
KB