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I have a quick question: Let's say I have a Lagrangian [itex]\mathcal{L} [/itex]. From Hamilton's principle I find a governing equation for my system, call it [itex] N\phi=0 [/itex] where N is some operator and [itex] \phi[/itex] represents the dependent variable of the system. If [itex]\mathcal{L} [/itex] has a particular symmetry, how does that (or does it at all) correspond to symmetries of the solution [itex] \phi [/itex]? ie does this symmetry map solutions to solutions?

Basically the essence of the question is this: Do the symmetries of the Lagrangian give us additional information about solutions to the governing equations?

Any help/references is appreciated.

Thanks,

Nick

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# Symmetries of Lagrangian and governing equations

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