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I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. In wikipedia I can see an equation that seems to give all the conserved quantities (wiki's article) but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain.

Qmechanic
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untreated_paramediensis_karnik
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    See my post http://physics.stackexchange.com/q/296555 – iiqof Dec 05 '16 at 09:27
  • @AngelJoaniquetTukiainen could you please comment on Ted Pudlik's answer below then? If I understand well, there exist a systematic way to obtain all conserved quantities of a system but it might involve tedious PDE systems to solve? And this is valid provided we do find the Lagrangian of the system. – untreated_paramediensis_karnik Dec 05 '16 at 20:15
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    I don't understand the request, what do you want me to coment?

    And yes, you have understand it correctly.

     – iiqof Dec 05 '16 at 20:56
  • @AngelJoaniquetTukianen Ted Pudlik claimed that there is no general algorithm to obtain the conserved quantities of a system while you claim that there is, under certain circumstances. I think it'd be nice if there some discussion with Ted occur. – untreated_paramediensis_karnik Dec 05 '16 at 21:43

2 Answers2

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There is no general algorithm for doing so, and even figuring out how many conserved quantities a system has can be difficult.

A famous example is the Toda lattice. When originally proposed by Toda in 1967, this model was believed to be chaotic. It was in fact proven to be integrable (to have too many conserved quantities to be chaotic) in 1974 by Henon. See Section 3.6 of Gutzwiller's Chaos in Classical and Quantum Mechanics for more details on this story.

Ted Pudlik
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  • Refering my related question: I think it clearly a hard problem, without a general algorithm, however it is not always impossible. Somehow proving it for an induvidual system, combined with the Noether theorem, would mean that all conserved quantities of that system were already found (i.e. the theory is "ready"). Doing that with the quantum mechanics would also mean a mathematical proof against Einstein's contra-argument (he did not accept the inherent uncertainity in the QM, and argued that it is because the theory is not ready). – peterh Nov 02 '18 at 20:01
  • What does it mean to have too many conserved quantities to be chaotic? Is there a maximum number of conserved quantities a system can have before it cannot be chaotic anymore? – HelloGoodbye Mar 10 '23 at 15:20
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Traditionally, human physicists figure out conserved quantities with extensive educated guesses and strong domain knowledge. Even for those cleverest minds (e.g. Henri Poincaré), it is hard to know whether all conserved quantities have been found.

However, recent progress in machine learning provides systematic approaches to this problem. Firstly, the number of conserved quantities can be obtained by formulating the original problem to a manifold learning problem. Secondly, all conserved quantities can be discovered numerically (parameterized as neural networks). Thirdly, some of them can even be discovered symbolically.

Here are a couple of recent papers I have written on the subject:

AI Poincare: Machine learning conservation laws from trajectories

AI Poincare 2.0: Machine learning conservation laws from differential equations

Valter Moretti
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Ziming Liu
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