the Rayleigh dissipation function

Standard Lagrangian mechanics assumes energy-conserving forces, but real systems have friction and drag. Rayleigh's trick is to package all the velocity-proportional damping into one extra scalar function, so that you can keep using the Euler-Lagrange machinery with only a small modification.

For damping forces proportional to velocity, you define F = (1/2) sum_ij c_ij q_dot_i q_dot_j, a quadratic form in the generalized velocities. The generalized dissipative force is then Q_i = -partial F / partial q_dot_i, and the equations of motion become d/dt (partial L / partial q_dot_i) - partial L / partial q_i + partial F / partial q_dot_i = 0. Physically, 2F equals the instantaneous rate at which mechanical energy is being dissipated, the power lost to friction.

The honest caveat is that this works only for forces linear in velocity, such as viscous Stokes drag or linear damping. It does not capture dry Coulomb friction, which has constant magnitude and the sign of the velocity, nor quadratic aerodynamic drag. It is genuinely an add-on that sits outside the pure variational principle, because dissipation cannot arise from an action that is stationary in the ordinary sense.