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In this Phys.SE question, one answer (by Ron Maimon) claims that one can make the assumption of a least action principle plausible using Liouville's Theorem as another starting point of the theory. The answer claims that conservation of information is equivalent to conservation of phase space volume (which is understandable and plausible to me), and follows from that that the time evolution of a system is given by a canonical transformation, and hence, by the canonical equations, in Hamilton formalism. I understand this part.

The answer then tries to provide an analog argument in the Lagrangian formalism, which I don't understand. It considers the space of all solutions in the configuration space as the phase space, but I quite don't understand the argument. Can anyone tell me how one can make plausible the derivation of the principle of least action, using the conservation of phase space volume (or something equivalent) in the Lagrangian formalism?

Edit: It would be nice if a possible answer could (as much as possible) make use of mathematics that are usually known to students of classical mechanics.

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  1. In the Lagrangian formalism on the space of on-shell paths in configuration space, there is an analog to Hamiltonian flow and Liouville's theorem in the Hamiltonian formalism in phase space, cf. e.g. Ref. 1 and Urs Schreiber's Phys.SE answer here.

  2. Example. For a Lagrangian of the form $L=\frac{m}{2} \dot{q}^2-V(q)$, one may show, using the Euler-Lagrange (EL) equation$^1$ $$\begin{align} m\ddot{q}~\approx~&-V^{\prime}(q)\cr ~\Downarrow~& \cr m\delta \ddot{q}~\approx~&-V^{\prime\prime}(q)\delta q,\end{align}\tag{A}$$ that the 2-form $$\omega~=~ m \delta \dot{q} \wedge \delta q\tag{B}$$ is a constant of motion (COM), $$\dot{\omega}~\stackrel{(B)}{=}~ m \delta \ddot{q}\wedge\delta q ~\stackrel{(A)}{\approx}~0,\tag{C}$$ cf. eqs. (14) & (15) in Ref. 1. Knowing that the corresponding Hamiltonian is just $H=\frac{p^2}{2m}+V(q)$, this is perhaps not so surprising.

  3. But generally, for an arbitrary Lagrangian $L(q,\dot{q},t)$, by using the EL equations $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^k}~\approx~\frac{\partial L}{\partial q^k} \tag{D}$$ and their consequences $$\begin{align} \frac{d}{dt}&\left(\delta q^j\frac{\partial^2L}{\partial q^j\partial\dot{q}^k}+\delta \dot{q}^j\frac{\partial^2L}{\partial \dot{q}^j\partial\dot{q}^k}\right)\cr ~\approx~&\delta q^j\frac{\partial^2 L}{\partial q^j\partial q^k}+\delta \dot{q}^j\frac{\partial^2 L}{\partial \dot{q}^j\partial q^k},\end{align} \tag{E}$$ one may show that the 2-form $$\begin{align}\omega ~=~&\delta\left(\frac{\partial L}{\partial \dot{q}^k}\right)\wedge\delta q^k\cr ~=~&\left( \delta q^j\frac{\partial^2L}{\partial q^j\partial \dot{q}^k} + \delta \dot{q}^j\frac{\partial^2L}{\partial \dot{q}^j\partial \dot{q}^k}\right)\wedge\delta q^k\end{align}\tag{F}$$ is a COM $$\begin{align}\dot{\omega}~\stackrel{(F)}{=}~&\frac{d}{dt}\left( \delta q^j\frac{\partial^2L}{\partial q^j\partial \dot{q}^k} + \delta \dot{q}^j\frac{\partial^2L}{\partial \dot{q}^j\partial \dot{q}^k}\right)\wedge\delta q^k \cr &+ \delta q^j\frac{\partial^2L}{\partial q^j\partial \dot{q}^k} \wedge\delta \dot{q}^k\cr ~\stackrel{(E)}{\approx}~&0.\end{align}\tag{G}$$ In this sense the volume/information is conserved also in the Lagrangian setting.

References:

  1. C. Crnkovic & E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.

  2. N. Reshetikhin, Lectures on quantization of gauge systems, arXiv:1008.1411; Subsection 3.2.1.

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$^1$ Here the $\approx$ symbol means equality modulo the EL equations, i.e. on-shell.

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  • $\begingroup$ I would just add a conceptual comment: the important point is that you use the Lagrangian/Hamiltonian to define what "information" is in the first place, and the construction is done precisely in a way so that this "information" is conserved. $\endgroup$
    – Void
    Dec 6 '19 at 12:45
  • $\begingroup$ @Void you so by introducing the meaningful measure for Information as the two-form $\delta q \delta \frac{\partial L}{\partial \dot{q}}$, Right? Does it also work the other way around? by introducing $\omega$ and requiring it to be conserved, are the Euler Lagrange equations the only ones that satisfy this requirement? $\endgroup$ Dec 8 '19 at 22:28
  • $\begingroup$ @Qmechanic Do I understand right that the reverse (starting from the "conservation" of a 2-form on on the space of solutions, and then arriving at the local Action functional) is covered in the paragraph "application to the inverse problem of the calculus of variations" on the phase-space-page of nlab? $\endgroup$ Dec 9 '19 at 10:35
  • $\begingroup$ @Qmechanic why would the two form $\omega$ correspond to information? $\endgroup$
    – Gonenc
    Dec 9 '19 at 16:13
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The space of all solutions of the Euler-Lagrange equations in the configuration space is equivalent to the phase space. Indeed, given an initial condition (consisting of a position and a velocity) there is a unique solution, and (assuming a fixed total mass) the velocity determines the momentum. Hence the points of phase space (i.e., pairs of position and momentum vectors) are in 1-1 correspondence with the set of all solutions.

As Ron Maimon points out, this description of the phase space has the advantage that it does not single out a specific time; one can get from the space of all solutions a traditional phase space description at an arbitrary time by considering the solution and its first derivative (time the mass) at that time.

This is a decisive advantage in covariant formulations of relativistic theories where space and time should appear on equal footing. Therefore the phase space as described in terms of solutions is also called the covariant phase space. The covariant phase space carries a natural symplectic form and an associated Poisson bracket called the Peierls bracket. See, e.g.,

  • C. Crnkovic, Symplectic geometry of the covariant phase space. Classical and Quantum Gravity, 5 (1988), 1557.

and the Physics SE threads here and here.

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  • $\begingroup$ Do I get right that the covariant phase space carries the natural symplectic form only if the solutions it consists stem from a local action functional? $\endgroup$ Dec 9 '19 at 7:42
  • $\begingroup$ @Quantumwhisp: The Lagrangian is usually assumed to be local in time. I don't know what happens in the nonlocal case. $\endgroup$ Dec 9 '19 at 9:40
  • $\begingroup$ The question was not meant towards the "local" part, but towards the "action functional" part. Do I understand right that the statement of your answer is "If there is a local action functional, it follows that the space of solutions carries a symplectic form"?, but it doesn't make a statement about the reverse, that the space of solutions can naturally be equipped with a symplectic form, and from that it follows that the Solutions must stem from an action principle? $\endgroup$ Dec 9 '19 at 10:40
  • $\begingroup$ @Quantumwhisp: I don't think a converse holds. There is a huge variety of possible dynamical systems, and only few of them are symplectic or derived from a variational principle. $\endgroup$ Dec 9 '19 at 14:21
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This probably isn’t as rigorous as the answer you’re looking for, but let me suggest that you could simply plot a small number of points in a simple phase space, with position in one spatial dimension on you x axis and momentum on the y axis. Start with (0,0), (0,1) and (1,0) to make it simple. If you begin with Liouville’s theorem, that the area defined by these points can not increase as it evolves, you can see that any arrangement that increases the area defined by your points will have to involve either momentum changing for no reason, or position changing, without momentum to account for it. Those would be the hallmarks of violation of principle of least action. Then you could generalize it and make it much more rigorous, of course. I could be mistaken, but I actually think Liouville’s theorem actually presupposes either conservation of momentum or principle of least action, or at least some minimal conserved quantity, otherwise, I don’t see how it could arrive at the conclusion about area staying the same.

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  • $\begingroup$ I don't find that a satisfying answer at all, since you didn't even give a hint on how the desired behaviour of the points should have anything to do with an action that is varied. $\endgroup$ Dec 24 '17 at 16:58

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