In reasoning about symmetries of dynamical systems usually there is an Legrangian $ L(p,q) $ and symmetry transformation $s' = f(s)$ where $s = p$ or $q$. If $f(s)$ represent continuous symmetry of the system, we use Lie Groups description and wrote $s'= f(s)= s + ðs + ...$ where $ðs$ is perturbation ( first, linear term in power expansion of $f(s)$ etc). Then $L(p',q') = L(p,q)$ and this lead us into well known Noether Theorem and integrals of motion.
- Supposing we have Lie semigroup ( or Lie monoid probably because of expansion around neutral element idea) instead and we try to use it to generalize Noether Theorem - is it possible? Are there any references for that idea?
- Is there any kind of introductory text to area of Lie semigroups or Lie monoids?
I imagine that it may lead to structure which recalls renormalization group equations - kind of parametrization between different systems sharing similar phase spaces or even different areas within the same phase space ( whilst when there is a true symmetry group there are, in contrast, separate subsurfaces of constant invariant of motion - a hypertori in a cause of integrable Hamiltonian system). For example I may speculate that averaging over (indistinguishable) initial conditions may lead to practical semimgroup "symmetry" realisation - it may be interesting in context of chaotic Hamiltonian dynamic. Let me be clear: I am not asking about physical interpretations here, amd I am not interssted in explanation why semigroup does not matter as symmetry of any kind of real system, but only about known facts about semigroup "symmetry" transformations when they occur in Hamiltonian dynamics. I am just curious if it gives something interesting.
It is cross-post from https://physics.stackexchange.com/questions/7856/noether-theorem-with-semigroup-of-symmetry-instead-of-group . It looks like physicist are not very interested in such question - probably it is not much physical - so I would like to ask it here - but of course if it does not fit - close it please.