Mathematical Format of Lagrangian vs Symmetries

[LarryS]

Most of the lectures that I have watched online say that a symmetry exists when the mathematical form of the Lagrangian does not change as a result of some transformation, like a local gauge change. But how does nature "know" the mathematical form of the Lagrangian? Obviously, I am missing the point.

Thanks in advance.

 

[Vanadium 50]

But how does nature "know" the mathematical form of the Lagrangian?

How does a thermos know to keep hoit liquids hot and cold liquids cold?

 

[LarryS]

How does a thermos know to keep hoit liquids hot and cold liquids cold?

Because thermoses are constructed of materials that are thermal insulators.

Actually, my question was more about the apparent relation of the mathematical form (mathematical syntax) of the Lagrangian expression to the physical system that it models. But, on second thought, I think the answer is that a transformation of a Lagrangian is considered a valid symmetry if the transformed Lagrangian leads to the same equation of motion. Comments?

 

[Vanadium 50]

The point is the thermos doesn't have to "know" anything to follow a mathematical description of its behavior - a description written down by people to describe behavior they have observed. This happens throughout physics.

 

[Nugatory]

say that a symmetry exists when the mathematical form of the Lagrangian does not change as a result of some transformation,

True, but you are emphasizing the wrong word. The important word here is "when" (as opposed to "because", which is how you're reading it). A symmetry exists when the Lagrangian has a particular transformation property, but that doesn't mean that nature had to know about the Lagrangian and introduce the symmetry to match the Lagrangian. It's the other way around; the symmetry is present in nature so the Lagrangian must reflect it.

 

[strangerep]

[...] on second thought, I think the answer is that a transformation of a Lagrangian is considered a valid symmetry if the transformed Lagrangian leads to the same equation of motion. Comments?

Indeed, symmetries of the equations of motion are more fundamental than symmetries of the Lagrangian. E.g., there are symmetries of the action (of which the Lagrangian is only the integrand) which involve changing the measure used in the action integral. [E.g., the "expansion" transformations in the Niederer-Schrödinger group, which generalizes the nonrelativistic Galilei group. Also some of the fractional-linear transformations that are symmetries of the free particle equation of motion.]

The equations of motion are what really matter. The Lagrangian/Hamiltonian formalisms are just ways of reformulating the theory in such a way that other features of the theory are easier to find and/or work with.

 

[vanhees71]

To be even more general: Not the action functional needs to be symmetric but only the variation of the action functional, because this is sufficient to make the equations of motion obeying the symmetry, and that's indeed all that counts.

Also it's clear that nature couldn't care less about how a Lagrangian looks. She is as she is, and we try to observe and describe her as good as we can doing physics, which leads to an amazingly simple and accurate description in terms of very esthetic mathematics. Why this is possible is very puzzling and not part of physics. It's what Wigner called the "the unreasonable effectiveness of mathematics in the natural sciences":

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
http://math.northwestern.edu/~theojf/FreshmanSeminar2014/Wigner1960.pdf