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How is Clairaut's Law modified to define geodesics in a twisted surface of revolution ( so not axi-symmetric)

$$ u \cos v, u \sin v , f(u) + T \, v ,$$

where T is a twisting constant? It appears that Clairaut's Law $u \sin(..,T) $ = constant could be defined. How to use the fact that geodesic or Gauss curvature remains zero/same even after twisting towards twist generalization?

EDIT1:

Assumed that $T$ can be placed in the derived pde like for conserved $K$s:

$$ \frac{u^3 f_1f_{11}- T^2}{[ u^2 (1+ f_1^2 )+ T^2 ]^2}= \frac{f_1f_{11}}{ u (1+ f_1^2)^2},$$

which simplifies to:

$$ \frac{T^2}{(1+f_1^2)}+ u ( 2 u + \frac{(1+f_1^2)}{f_1f_{11} } )= 0$$

where partial differentiation w.r.t. $v$ does not occur.

When T=0, the following should reduce to Clairaut:

$$ u + \frac{(1+f_1^2)}{ 2 f_1f_{11} } = 0 $$

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How is Clairaut's Law modified to define geodesics in a twisted surface of revolution [...]?

In a sense, "not at all". Clairaut's relation comes from intrinsic geometry, assuming a metric has the form $$ g = E(u)\, du^{2} + G(u)\, dv^{2}. \tag{1} $$ The trick, from this perspective, is to find an isometric embedding of (1) into Euclidean $3$-space in a manner equivariant with respect to a helical group of isometries (i.e., so that intrinsic translation in $v$ is the same as extrinsic rotation-and-twisting about an axis).

The parametrization $$ X(u, v) = (u\cos v, u\sin v, f(u) + Tv) $$ is inconvenient for studying geodesics, since the coordinate curves are non-orthogonal (i.e., $F = Tf'(u) \neq 0$). Instead of taking the "profile" curve to lie in a plane containing the "axis" of the surface, however, you can take the profile to be a "helix" along the surface that stays everywhere orthogonal to the $v$-coordinate curves.

Geometrically, the assertion of the preceding paragraph is fairly clear. Analytically, for a fixed geometric surface, finding a suitable helix entails solving an ODE that controls the "retrograde winding" of the profile. The resulting parametrization is Clairaut, i.e., of the form (1), so its geodesics are determined by Clairaut's relation, and can be parametrized using the equivariant embedding.

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