In many dynamic systems in classical physics, as well as quantum mechanics, the equation of motion can be derived from a variational principle (VP), i.e. minimizing an action integral of some sort.
I am wondering what the structure is that is already built into a system by merely assuming that it is derived from a VP?
It seems that stationarity of the action under variation must imply some kind of symmetry and henceforth a conserved quantity. What is it?
Since VP often leads to a Hamilton formulation that implies Liouville’s theorem, could it be that the conserved quantity is the conservation of phase space? (In quantum mechanics it would correspond to unitarity) . Hence information can never get lost in systems where the dynamic is derived from a VP?