= Small-slope particle motion on a height graph
{title2=$\ddot{\mathbf r}\simeq-g\nabla h$}
A particle constrained to the graph $z=h(x,y)$ has <kinetic energy> $m[|\dot{\mathbf r}|^2+(\nabla h\cdot\dot{\mathbf r})^2]/2$ and <potential energy> $mgh$. For a gently varying graph and velocities for which its geometric kinetic correction is higher order, the leading <Euler-Lagrange equations> are the displayed planar gradient-force equations. Small slope alone at arbitrarily large speed or curvature does not justify discarding the normal inertial acceleration. For a radial height, the leading planar <angular momentum> is conserved, allowing an <effective potential> reduction.
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