Do dynamic systems that are based on a variational principle imply a conservation law? 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?
 A: *

*Concerning OP's first question: One can argue that a notion of volume/information is conserved in the Lagrangian formulation, cf. e.g. my Phys.SE answer here.


*Concerning OP's last question: The fact that the Hamiltonian formulation in phase space has a greater symmetry (symplectomorphism symmetry, Liouville's theorem) than the corresponding Lagrangian formulation in the configuration space is mostly due to the use of twice as many variables (arranged in a balanced manner). See also e.g. this, this, this, this & this related Phys.SE posts.
Note that if one is allowed to introduce new variables, they often bring new symmetries.
Example: If we have an action $S[x]$ that depends on the variable $x$ and we introduce a new variable $y$, then a transformation $y\to y^{\prime}=f[y]$ is a trivial symmetry of the action $S[x]$. (This example is used in Srednicki, QFT, chapter 71, as a model for gauge symmetry.)
A: No, there is no "general conservation law" like what you imagine. The reason is simple: a variational principle is just that, a variational principle. It need not have anything to do with physics at all. It could be something as simple as finding the vertex of a parabola. What is considered a dynamic system is a matter of taste because time is just an arbitrary variable to mathematics.
Suppose you have given a function $\phi(t)$ and you want to find the trajectory $x(t)$ that minimizes the action
$$S=\int \left(\frac{dx}{dt}-\phi(t) \right)^2 dt$$
The result is of course trivial:
$$x(t)=\int^t \phi(\tau)d\tau +C$$
There is no way to find any conservation law, because $\phi(t)$ is completely arbitrary.
But if $\phi$ does not depend on time, the Lagrangian becomes explicitely time-invariant, which gives you something similar to energy conservation according to Noether's theorem.
