# 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?

1. 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.

2. 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.)

• Is the Liouville's thm of Hamilton systems something deep, or rather something shallow ( 'twice as many variables)? I thought the former:'Liouville's theorem can be thought of as conservation of information in classical mechanics'.see : (physicstravelguide.com/theorems/liouvilles_theorem) Commented Apr 5, 2021 at 7:14
• It seemed to me that the 'minus sign' in the symplectic form was the most crucial aspect in deriving Liouville's thm. Hence: Action integral --- variation---> Euler Lagrange ---- Legendre transf ---> Hamilton formalism --- Liouville thm--> conservation of information. Hence models based on variation of an action 'automatically' preserve information Commented Apr 5, 2021 at 7:52
• I updated the answer. Commented Apr 5, 2021 at 8:17
• thx you for your illuminating links (: Commented Apr 5, 2021 at 13:35

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.

• Minor comment to the example (v3): $x$ is a cyclic variable, so momentum is a constant of motion... Commented Apr 5, 2021 at 7:24
• @Qmechanic: I tried to update my answer accordingly. But then I realized that although the solution I have given ($\frac{dx}{dt}-\phi=0$) is obviously the absolute minimum of the action, the Lagrange equations only yield $\frac{dx}{dt}-\phi=const.$ (which is the generalized momentum conservation, you have mentioned). What am I missing? Commented Apr 5, 2021 at 10:12
• Your example fits into a Hamiltonian framework for $H= 1/4 \ p^2-p \ \Phi$ where $p=2(\dot q -\Phi)$ hence one should get Liouville's thm and hence conservation of phase space, or am I mistaken? Commented Apr 5, 2021 at 13:27
• If one were to consider $\ddot q = -k \ q-\beta \dot q$ on the other hand, there seems no action and indeed Liouville thm does not seem to hold. Would you agree that in the example you gave, preservation of phase space holds? Commented Apr 5, 2021 at 13:34
• @user3072048: as Qmechanic has noted, the example is not yet suitable because it has conserved momentum. I will fix this as soon as I find more time. As to phase space symmetry, see Qmechanic's answer (part 2). Commented Apr 5, 2021 at 13:37