Justification of the Least Action Principle using conservation of information

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.