= Jacobi energy in a shearing sheet
{c}
{title2=$\varepsilon_J$}
The time-independent <particle Lagrangian in a shearing sheet> has conserved rotating-frame energy
$$
\varepsilon_J=\sum_i\dot q_i\frac{\partial L_2}{\partial\dot q_i}-L_2
=\frac12(\dot x^2+\dot y^2+\dot z^2)-q\Omega_0^2x^2+\frac12\Omega_z^2z^2.
$$
The velocity-linear <Coriolis acceleration> term cancels from this expression. Up to the reference-orbit constant, it is the second-order expansion of inertial <specific orbital energy> minus $\Omega_0$ times inertial <specific angular momentum>. Its negative radial tidal term allows <inelastic collisions> to lower the total energy while increasing the radial extent of a ring.
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