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 particles as

The force of friction is negative the velocity gradient of the dissipation function, . The function is half the rate at which energy is being dissipated by the system through friction.