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Many of important equations in physics satisfy the least action principle, namely they are Euler-Lagrange equations for some Lagrangian. On the other hand it is known in mathematics that not all differential equations (both ODE and PDE) are Euler-Lagrange.

Are there physically important (!) equations which are not Euler-Lagrange?

Qmechanic
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MKO
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  • This post (v2) seems like a list question. – Qmechanic Jan 17 '18 at 14:33
  • Possible duplicates: https://physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Jan 17 '18 at 14:38
  • Are you asking generally about equations that cannot be written in terms of an action principle or about equations that a physicist might expect or hope to be able to write in terms of an action principle but can't? (I would guess the latter, as I expect the former has many 'trivial' examples) – By Symmetry Jan 17 '18 at 14:45
  • @BySymmetry I am asking about equations about which it is known or expected that they cannot be written in terms of action principle. But also I would interested in examples when such an interpretation is unknown although it is expected. – MKO Jan 17 '18 at 17:13

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"Physically important" depends on who you ask, but Type IIB supergravity theory in 10 dimensions does not have a simple action principle that is Lorentz-invariant. The standard reference for this is

Neil Marcus & John H. Schwarz, "Field theories that have no manifestly Lorentz-invariant formulation." Physics Letters B115, 111–114 (1982).

However, it is possible to construct an action for this theory by introducing auxiliary fields and/or breaking Lorentz invariance. See the following reference for more details and some historical references:

Ashoke Sen, "Covariant action for type IIB supergravity." Journal of High Energy Physics 2016:17 (2016).

Michael Seifert
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