Rayleigh dissipation function

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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.