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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?

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2 Answers 2

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  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.)

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  • $\begingroup$ 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) $\endgroup$ Commented Apr 5, 2021 at 7:14
  • $\begingroup$ 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 $\endgroup$ Commented Apr 5, 2021 at 7:52
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    Commented Apr 5, 2021 at 8:17
  • $\begingroup$ thx you for your illuminating links (: $\endgroup$ Commented Apr 5, 2021 at 13:35
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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.

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    $\begingroup$ Minor comment to the example (v3): $x$ is a cyclic variable, so momentum is a constant of motion... $\endgroup$
    – Qmechanic
    Commented Apr 5, 2021 at 7:24
  • $\begingroup$ @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? $\endgroup$
    – oliver
    Commented Apr 5, 2021 at 10:12
  • $\begingroup$ 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? $\endgroup$ Commented Apr 5, 2021 at 13:27
  • $\begingroup$ 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? $\endgroup$ Commented Apr 5, 2021 at 13:34
  • $\begingroup$ @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). $\endgroup$
    – oliver
    Commented Apr 5, 2021 at 13:37

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