In dimension 3, the equation for unparametrized conformal geodesics is variational.
Krupka, The Vainberg–Tonti Lagrangian and the Euler–Lagrange mapping, Differential Geometric Methods in Mechanics and Field Theory, Eds: F
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Variationality of conformal geodesics in dimension 3
In dimension 3, the equation for unparametrized conformal geodesics is variational.