= Particle Lagrangian in a shearing sheet
{title2=$L_2$}
Expand a unit-mass particle's <Lagrangian> about an axisymmetric <circular orbit> of radius $r_0$ and frequency $\Omega_0$, using $r=r_0+x$ and $\varphi=\Omega_0t+y/r_0$. Circular-orbit balance removes the linear radial term; <total-time-derivative invariance of a Lagrangian> removes $r_0\Omega_0\dot y$. To second order,
$$
L_2=\frac12(\dot x^2+\dot y^2+\dot z^2)+2\Omega_0x\dot y-\Phi_t,
\qquad
\Phi_t=-q\Omega_0^2x^2+\frac12\Omega_z^2z^2.
$$
Here $q$ is the <orbital shear parameter> and $\Omega_z$ is the <vertical epicyclic frequency>. The <Euler-Lagrange equations> retain the local <Coriolis acceleration> and tidal gravity.
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