If we take the Euler-Lagrange equation and treat it as a linear operator acting on a Lagrangian,
(∂/∂q - d/dt ∂/∂q*)L = 0
then it appears that the Lagrangian for a system is in the null space for that operator. Is this point of view used anywhere that you know of? Is there meaning in finding the eigenfunctions/eigenvalues of this operator?
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