Can metrics or spacetimes which globally violate the laws of energy conservation be described by Lagrangians?

Question

This question is a follow-up one from a short discussion I had in the comments of an answer of my previous question

There, I asked whether cellular automata can be described by Lagrangians (or Hamiltonians) and the answer said that they generally wouldn't be described that way because in general cellular automata do not respect any conservation laws or symmetries, so they will not be representable by Lagrangians.

However, then, concerning metrics and spacetimes of the universe which violate symmetries like global translational invariance and do globally violate the law of conservation of energy cannot be described by Lagrangians?

Answer 1

In a sense, energy is not globally conserved in general relativity (particularly in cosmology). The reason is that the background is dynamical. As the universe expands, objects within it tend to lose energy and eventually follow the cosmic rest frame determined by the expansion. You can look up cosmological redshift, Hubble flow, or Hubble drag.

As an explicit example, consider the Lagrangian for a Klein–Gordon scalar field in a FLRW background. You get $$S = \frac{1}{2} \int a(\eta)^2 \left((\varphi')^2 - (\nabla\varphi)^2\right)\mathrm{d}^3x \mathrm{d}\eta,$$ where $\eta$ denotes conformal time and a prime denotes a derivative with respect to conformal time. The time-dependence in the scale factor $a(\eta)$ can make the scalar field lose energy over time.

Answer 2

You did not have to go so far. Even just in introductory analytical mechanics, you could write down a Lagrangian that explicitly depended upon time (and is thus not Lorentz invariant) and it would violate energy conservation. You can do the same violent alterations to a spacetime Lagrangian and obtain the lack of energy conservation too.

Answer 3

A system described in terms of a Lagrangian is likely better phrased as a system which can be captured using Hamilton's principle that the true flow through time of a system is described as a stationary path given a Lagrangian. For Netwonian mechanics, it is easy to prove that all physical systems have a Lagrangian in the form of kinetic energy minus potential energy. If an action exists such that Hamilton's principle applies, it implies the existence of a Lagrangian. However, it is not guaranteed that such a function exists. Indeed, any non-conservative system does not have a Lagrangian in its truest sense (Galley did come up with a variation on the concept which captures a large range of non-conservative systems).

You mention cellular automata. Cellular automata are discrete systems. The entire concept of calculus of variations behind Hamilton's principle and thus Lagrangian mechanics assumes a continuous state space. There are other mathematical models which are used for discrete spaces.

Answer 4

Hamiltonian is not necessarily giving energy. In general, a time-independent Hamiltonian has to be conserved, while energy need not be. The simplest example is perhaps the harmonic oscillator; for that, see e.g. my answers here:

When is the Hamiltonian of a system not equal to its total energy?

Intuitive explanation for why time symmetry implies conservation of energy?

So yes, some systems that do not conserve energy can be described by Lagrangians/Hamiltonians. However, cellular automaton is a discrete step system; Lagrangians and Hamiltonians were meant and apply well for continuously evolving systems.

I don't know if there is a re-interpretation of Lagrangian/Hamiltonian description for discrete state systems. Maybe if the update rule of the automaton is augmented/completed into a continuous evolution in time. But then there would be new states "in between" two classical states of the automaton, so it would something very different.

Discrete computation is conceptually very different from evolution of a Lagrangian/Hamiltonian physical system, similar to how the set of integer numbers is very different from the set of real numbers. Discrete state system has a finite number of details; but continuous system has infinitely fine detail and there is no limit to how small a change can be made.