# Rayleigh dissipation function

In physics, the Rayleigh dissipation function, named for Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. It is defined for a system of $N$ particles as
$F = \frac{1}{2} \sum_{i=1}^N ( k_x v_{i,x}^2 + k_y v_{i,y}^2 + k_z v_{i,z}^2 ).$
The force of friction is negative the velocity gradient of the dissipation function, $\bold{F}_f = -\nabla_\bold{v} F$. The function is half the rate at which energy is being dissipated by the system through friction.